ds05 abstracts · 2018. 5. 25. · short-strand dna renaturation by an active micro-mixer...

76 DS05 Abstracts

CP1

Short-Strand Dna Renaturation by An ActiveMicro-Mixer

Understanding renaturation of short (40-50 mer long) ss-DNA strands in flow is of interest from both the theoreticalperspective and from that of experimentalists studying me-chanical properties of DNA. Since the size of these strandsis less that the persistence length of ssDNA, we model thestrands as rods. Assuming that two rods combine to formdsDNA if they are close together and are closely aligned,and considering the dilute limit, we model the process as areaction-diffusion-advection system in position-orientationspace where the reaction term has a particularly simpleform. The model contains avenues for control in the formof the advective velocity fields and applied external poten-tials. We employ the model to study the enhancement ofreaction rate by an active micro-mixer consisting of chan-nel that is 200 x 100 microns in cross section and is a fewhundred microns in length. Two streams containing eachtype of ssDNA are introduced at the inlet and are agitatedby fluid in side-channels (50microns wide) before reachingthe outlet. We perform 2-D as well as 3-D analysis usinganalytical expressions for the underlying velocity field. Wealso consider another process, where streams of dye anddsDNA are introduced into the micro-mixer. The reactionconsists of the dye attaching itself between the strands ofthe DNA, subsequent to which the dye fluoresces and theintensity of fluorescence can thus indicate amount of bounddye. This process is treated as a special case of the samemodel.

Igor Mezic, Thomas JohnUniversity of California, Santa [email protected],[email protected]

CP1

A Dynamical Systems Approach to ReconstructingRepetitive Dna

Mathematical methods in biology, especially regardinganalysis of DNA sequence information has led to majorresults in recent years. Still, the task of determining theDNA sequence of an organism, genome assembly, remainsan expensive and lengthy process. Repetitive regions ofDNA are especially difficult to assemble. We develop amethod inspired by ideas from Symbolic Dynamical Sys-tems to reconstruct repetitive DNA. We the validity of ourmethod by analyzing two different assemblies of Drosophilamelanogaster (fruit fly). We conclude by describing waysour method can be used to improve genome assembly.

Suzanne SindiUniversity of Maryland, College [email protected]

CP1

Mesoscale Modeling of Protein-Dna Complexes

DNA transcription begins with the formation of a complexof DNA and proteins. Recently developed base-pair leveltheory of DNA elasticity enables construction of structuraland dynamical models of such complexes that yield newinformation about the role of the promoter sequence in themechanism of gene regulation. Presented will be applica-tions to two complexes important for the regulation of theLac operon in E. coli: the LacR-DNA promoter complex

and the CAP-dependent activation complex.

David SwigonDepartment of MathematicsUniversity of [email protected]

CP2

Patched Heteroclinic Orbits in a Family of ScalarWave Equations

We analyze traveling wave solutions of the following familyof scalar wave equations:

ut + uux = −γα2G ∗(u2x

)x, (1)

for γ > 0, where G is the Green’s function of the Helmholtzoperator Hu = u−α2uxx. We find a novel class of patchedheteroclinic orbits which appear to cross a line of singu-larities in the (u,u’) phase plane. As γ increases past thecritical value γ = 1, we observe a qualitative change in thebehavior of trajectories near the singularity line. Theseorbits correspond to shock-like traveling wave solutions of(1), which in the zero-α limit, are global weak solutions ofBurgers’ equation.

Razvan C. FetecauStanford UniversityDepartment of [email protected]

Harish S. BhatCalifornia Institute of TechnologyControl and Dynamical [email protected]

CP2

Time Simulation-Based Bifurcation Analysis ofWaves

In Nonlinearity 16:1257-1275, 2003, we presented a sym-metry reduction of a PDE which can be used to ”freeze”waves and self-similar solutions. In this presentation, wewill review strategies based on time simulation to com-pute wave solutions, some based on the above symmetry-reduced PDE. We will also argue that a generalized eigen-value problem is a better way to determine the stabilitysince it can filter out the neutral eigenvalues.

Clarence RowleyPrinceton UniversityDepartment of Mechanical and Aerospace [email protected]

Ioannis KevrekidisDept. of Chemical EngineeringPrinceton [email protected]

Kurt LustUniversity of GroningenInstitute of Mathematics and Computing [email protected]

CP2

Traveling Waves for Differential-Difference Equa-tions with Inhomogeneous Diffusion

Traveling wave solutions for lattice differential equations

DS05 Abstracts 77

are defined by boundary value problems with advances anddelays on an unbounded domain. Fourier transform tech-niques and Jacobi operator theory allow one to obtain an-alytic results for a problem of this type with a diffusioncoefficient that is varied in some interval on the lattice.Of particular interest is the issue of propagation failureof these traveling waves depending on the inhomogeneousdiffusion.

Tony R. HumphriesMcGill UniversityMathematics & [email protected]

Brian E. MooreDepartment of Mathematics and StatisticsMcGill [email protected]

Erik Van VleckDepartment of MathematicsUniversity of [email protected]

CP3

Feedback Linearization of Chemostats

The chemostat is a rare example of a class of biological sys-tems that has real experimental and industrial applicationsand also admits a modeling paradigm that yields rigor-ously analyzable mathematical models. We apply the dif-ferential geometry-based methodology of nonlinear controlto mathematical models of chemostats. We show that byproperly choosing control parameters these models can bemade equivalent to linear dynamical systems and suitablychosen control objectives can be met via linear methods.We stress that this is not an approximation, but actuallyamounts to an analytic equivalence of the full nonlinearsystem with controls to a controllable linear system.

Mary BallykNew Mexico State U.Dept of Mathematical [email protected]

Ernest BaranyNew Mexico State UniversityDept of Mathematical [email protected]

CP3

Geometric Control and Chaotic Synchronization

We present a new parameter estimation procedure for non-linear systems. Such technique is based on the synchroniza-tion between the model and the system whose unknownparameter is wanted. Synchronization is accomplished bycontrolling the model to make it follow the system. We usegeometric nonlinear control techniques to design the con-trol system. These techniques allow us to derive necessaryand sufficient conditions for synchronization and hence forproper parameter estimation. As an example, this proce-dure is used to estimate a parameter of an example servingas a model.

Ubiratan FreitasLAC - Laboratory for Computing and AppliedMathematicsINPE - Brazilian Institute for Space Research

[email protected]

Elbert E. MacauLAC - Laboratory for Computing and AppliedMathematicsINPE - Brazilian Institute for Space [email protected]

Celso GrebogiInstituto de Fisica - IFUniversidade de Sao Paulo/ [email protected]

CP3

Dynamics of Adaptive Delayed-Feedback ControlSystems

First, we derive an adaptive control method for discretetime maps, by extending delayed-feedback controls alreadyproposed. Then, we study dynamics of adaptive controlsystems. In particular, we apply our method to the Henonmap, and numerically show the following two character-istics: Power law decay of distribution of control times.Almost zero finite-time Lyapunov exponent. With an an-alytical treatment, we show the simplest control systembecomes neutrally stable, which well explains these char-acteristics.

Asaki SaitoFuture University - [email protected]

CP4

The Use of Iterated Function Systems in Cryptog-raphy and Steganography

Iterated Function Systems (IFS)has found its applicationsin many areas of mathematics and our lives. In this talkwe show that one can hide data such as CryptographyKeys and messages in the attarctors of IFS such that theintended observers or recipients can reterive the messagefrom the atractor. The algorithm along with several exam-ples will be disscussed.

Mohammad KhadiviJackson State UniversityDepartment of [email protected]

CP4

Bounded Nonwandering Sets for Polynomial Maps

In this paper, we consider a class of polynomial mapson Rm or Cm which is defined by the assumption thatthe delay equations induced by the maps have leadingmonomials of single variable. We show that for any mapfrom this class, the nonwandering set is bounded whilefor all unbounded orbits, some kind of monotonicity takesplace. The class under consideration is proved to contain inparticular the generalized Henon maps and the Arneodo-Coullet-Tresser maps.

Mikhail MalkinDepartment of Mathematics, Nizhny Novgorod StateUniversityNizhny Novgorod, [email protected]

78 DS05 Abstracts

Ming-Chia LiDept. of Math., National Changhua University [email protected]

CP5

Turbulent Spin Dynamics under the Joint Action ofthe Distant Dipolar Field and Radiation Damping

Nonlinear evolution of the microscopic magnetization andexperimentally measured net magnetization in high-fieldsolution magnetic resonance is analyzed. These dynam-ics arise from the joint action of the distant dipolar fieldand radiation damping and have been recently used to en-hance sensitivity and contrast in magnetic resonance spec-troscopy and imaging. The dynamics beyond the initialtransient regime are shown to reach a turbulent phase. Theobserved turbulence may be a manifestation of spatiotem-poral chaos.

Susie Huang, Yung-Ya LinDepartment of Chemistry and BiochemistryUniversity of California, Los [email protected], [email protected]

Sandip DattaDepartment of Chemistry and BiochemistryUniversity of California, Los Angeles (UCLA)[email protected]

CP6

On Some Zero-Preserving Iso-Spectral Flows

I am studying the following general problem: Given a’structured’ symmetric matrix, (efficiently) find its eigen-values. In particular, I am studying flows in the spaceof symmetric matrices which preserve eigenvalues (that is,are ’iso-spectral’) and preserve structure. The Toda flowis an example; this flow is iso-spectral and preserves tridi-agonal structure. (This flow is closely related to the QRalgorithm.) Let A and B be positive semi-definite linearoperators on an inner product space V. Let Inv denote theMoore-Penrose inverse operation. Let A!B := A Inv(A+B)B. (This linear operator is called the ’quasi-projection’,’harmonic mean’ or ’parallel sum’ of A and B.) It is knownthat the range of A!B is equal to the intersection of therange of A and the range of B. We can use this operationto design structure preserving iso-spectral flows. I shall de-scribe some iso-spectral flows which preserve sparsity - thatis, preserve zero entries. I shall show that these flows con-verge. I conjecture that these flows converge to diagonalmatrices. Numerical experiments support this conjecture.

Kenneth DriesselColorado State [email protected]

CP6

A Class of Integrable Dynamical Systems and TheirConnection to One-Dimensional Inverse Problem

We present a large class of exact solutions to A-equationintroduced by Barry Simon and reveal a new class of arbi-trarily large systems of nonlinear ordinary differential equa-tions, which we show to be C-integrable in the sense of F.Calogero. Integration scheme is proposed and the approach

is illustrated in several examples.

Yilian ZhangUniversity of South Carolina [email protected]

Adrian NachmanUniversity of [email protected]

CP7

Basin Hopping in a Gearbox Model

We analyse the consequences of noise in a model for rat-tling in a single-stage gearbox with a bachlash. One ofobservable effects is the basin hopping, i. e., the switch-ing between basins of different attractors. This is a conse-quence of the intertwined nature of the basins of attractionor the presence of chaotic transients.

Silvio SouzaInstitute of PhysicsUniversity of Sao [email protected]

Antonio BatistaDepartment of MathematicsState University of Ponta [email protected]

Ibere L. CaldasInstitute of PhysicsUniversity of Sao [email protected]

Ricardo VianaPhysics DepartmentFederal University of [email protected]

CP7

Period Doubling in An Interrupted Machining Pro-cess

In this lecture a nonlinear delay-differential equation modelof high-speed milling is studied. The model incorporatesthe discontinuous ’fly-over’ effect, when the tool leaves theworkpiece due to high amplitude vibrations. First, we in-troduce a numerical method to analyse periodic solutionsin the periodic delay equation model. We use pseudo-arclength continuation technique to follow the bifurcationcurves of period-two orbits. To check the numerical find-ings we compute the sense of the period doubling bifurca-tion analytically.

Robert SzalaiMassachusetts Institute of [email protected]

CP7

Gear Rattle in the Large Stiffness Low DampingLimit

We examine models of gear rattle that take the form of pe-riodically forced ODE oscillators which are linear up to theinclusion of a backlash term that models the meshing forcebetween gears in terms of their relative rotational displace-ment. We use piecewise-smooth dynamical systems theory

DS05 Abstracts 79

to develop bifurcation diagrams that demonstrate the co-existence of a variety of stable rattling behaviors. Indus-trial applications include the control of noise and vibrationproblems in vacuum pumps.

Eddie WilsonUniversity of Bristol, [email protected]

CP8

Real-Time Construction of Optimized Predictorsfrom Data Streams

A new approach to the construction and optimization oflocal models is proposed in the context of data streams,that is, unlimited sources of data where retaining and pro-cessing all observations is impractical. A learning data setof limited size optimized in terms of predictive power isextracted. Real-time revision of the learning set allows se-lective coverage of regions in state space which contributemost to reconstructing the underlying dynamical system.

Leonard SmithOCIAMUniversity of [email protected]

Frank KwasniokUniversity of Oldenburg, [email protected]

CP8

Testing Causality: Coupling Asymmetry from Bi-variate Series Using Surrogate Data

Bivariate time series measured in coupled dynamical sys-tems can be used to infer asymmetry of coupling and thusthe causality in evolution of the interacting (sub)systems.We compare the size and power of asymmetry tests us-ing various types of surrogate data. Real-world applica-tions are presented in establishing causality relations inclimate evolution (relations between near-decadal oscilla-tory modes of long-term temperature records and NorthAtlantic Oscillation index) and in the electrophysiologicaldata capturing brain-cardiac-respiratory interactions.

Milan PalusAcademy of Sciences of the Czech RepublicInstitute of Computer [email protected]

CP8

On Model Reduction and Nonlinear Parameter Es-timation of Multivariate Time-Series

Given a multi-variate time-serie, which is obtained fromsome unknown ODE, we construct an approximating ODEthat well constructs the data. In this talk, we will introducea technique to fit experimental data sets by global modelingand parameter estimation. Also we will give the proof ofthe convergence of the parameter matrix in details. Somenumerical results and prospective work will be given in theend.

Chen YaoBox 5817 Clarkson UniversityPotsdam, NY [email protected]

Erik BolltClarkson [email protected]

CP9

Virtual Rigid Bodies in the Circular, Re-stricted Three-Body Problem: Dynamically-Natural Spacecraft Formations

The circular, restricted three-body problem provides aricher dynamical environment than the two-body problem.There has been a considerable amount of work in recentdecades, which takes advantage of equilibrium, stability,instability, and chaos to develop low energy trajectories.Periodic and quasi-period orbits in this realm are exam-ined to determine if and what types of spacecraft forma-tions can be maintained with little to no propulsion, so asto create virtual rigid bodies in space.

Ralph R. BasilioUniversity of Southern [email protected]

Paul K. NewtonUniv Southern CaliforniaDept of Aerospace [email protected]

CP9

Homoclinic Points Near Resonances in the Re-stricted Three-Body Problem

Resonance between the periods of planets, satellites, andother objects is a feature of celestial mechanics. Amongthe many extra-solar planets discovered recently, many res-onances have been observed. This talk will consider reso-nant periodic motions in the restricted three-body problemwith the aim of gaining an understanding of instabilitiescaused by resonance. I will outline a demonstration of theexistence of homoclinic points near resonances.

Divakar ViswanathUniversity of [email protected]

CP9

Computation of Low Energy Earth-to-Moon Trans-fers and Their Control by a Novel Method

We construct a spacecraft transfer with low cost and mod-erate flight time from the Earth to the Moon. The mo-tion of the spacecraft is modeled by the planar circular re-stricted three-body problem including a perturbation dueto the solar gravitation. Our approach is to reduce compu-tation of optimal transfers to a nonlinear boundary valueproblem. Using a computer software called AUTO, we solveit and continue its solutions numerically to obtain the op-timal transfers. Moreover, we show that the optimal trans-fers are unstable and can be stabilized by a novel controlmethod.

Kazuyuki YagasakiGifu UniversityDepartment of Mechanical and Systems [email protected]

80 DS05 Abstracts

CP10

Optimal Routing of a Sailboat in Steady Winds

We consider a family of novel, insightful, but yet tractableproblems regarding the optimal routing of a sailboat inthe plane, such as a racing yacht, in steady (i.e. time-invariant) winds. For the case of non-constant, but smoothwind fields, the minimum-time problem takes on the formof a Zermelo problem, resulting in a 2-point boundary-value problem generated from application of the calculusof variations and requiring a single iteration of the initialboat heading angle.

Michael P. HennesseySchool of EngineeringUniversity of St. [email protected]

Jeffrey JalkioUniversity of St. [email protected]

CP10

Early Detection of Anoxic Crises in Coastal La-goons

An important and recurrent problem in coastal lagoons isanoxia. Anoxic crises occur mainly in summer when tem-peratures are high and are triggered off by organic mat-ter decomposition. The early detection of anoxic crisesis therefore an important aspect in the management ofcoastal lagoons.In this work techniques from non-linear dy-namical systems theory have been applied to oxygen con-centration for estimating in advance the occurrence of ananoxic crises.

Dimitar MarinovEuropean CommissionJoint Research [email protected]

Francesca SomaEuropean Commission,Joint Research [email protected]

Eugenio GutierrezEuropean CommissionJoint Research [email protected]

Jose Manuel ZaldivarEuropean Commission, Joint Research Centre, [email protected]

CP11

Regular and Irregular Cycling Near a HeteroclinicNetwork

Heteroclinic networks (collections of robust heteroclinic cy-cles sharing common equilibria) appear naturally in sym-metric dynamical systems, yet little in general is knownabout the behaviour of typical nearby trajectories. Thistalk discusses a specific example of a heteroclinic networkthat displays a variety of interesting dynamics. In par-ticular, trajectories are observed to settle into a series ofexcursions around different parts of the network that wecall ‘cycling sub-cycles’. Cycling patterns displaying dif-

ferent numbers of loops around the individual componentcycles can be stable for the same parameter values, as cancombinations of regular and irregular (aperiodic) cycling.Analytical results for the regular cycling behaviour agreewell with numerical simulations, and explain, at least par-tially, the occurrence of the irregular cycling.

Claire M. Postlethwaite, Jonathan DawesDAMTP, University of [email protected],[email protected]

CP11

Robust Heteroclinic Cycles in the 1D ComplexGinzburg-Landau Equation

A new analysis is undertaken to explain numerical resultsshowing previously unobserved stable heteroclinic cyclesin large parameter regions of the 1D complex Ginzburg-Landau equation on a periodic domain. These cyclesconnect different spatially and temporally inhomogeneoustime-periodic solutions. We delineate regions of existenceand stability and indicate rich dynamics when the cyclesbecome unstable, including Shilni’kov-Hopf and blow-outbifurcations.


David LloydUniversity of [email protected]

Alan ChampneysUniversity of BristolDept. of Engineering [email protected]

CP11

A Codimension Two Resonant Bifurcation from aHeteroclinic Cycle with Complex Eigenvalues

Robust heteroclinic cycles in systems with symmetry canundergo a variety of bifurcations. The resonant bifurcationis usually associated with the birth or death of a nearbyperiodic orbit, and as such can occur in a supercritical orsubcritical manner. We investigate the degenerate case be-tween these two for a specific heteroclinic cycle with com-plex eigenvalues. A complex but ordered structure of fur-ther bifurcations of periodic orbits is found around thiscodimension two point.

Claire M. Postlethwaite, Jonathan DawesDAMTP, University of [email protected],[email protected]

CP12

Bifurcation Analysis of a Differential EquationsModel for Mutualism

We develop from basic principles a two-species differen-tial equations model which exhibits mutualistic populationinteractions. We vary the intrinsic growth rates of the pop-ulations and perform a bifurcation analysis. The bifurca-tions of primary ecological interest are those which allowthe population(s) to survive, but more interesting bifur-

DS05 Abstracts 81

cation phenomena are also observed. The model reducesto the familiar Lotka-Volterra model locally, but is morerealistic globally in the case where mutualist interaction isstrong.

Bruce B. PeckhamUniv. of Minnesota, DuluthDept. of Mathematics and [email protected]

Wendy GravesRainy River Community [email protected]

John PastorUniversity of Minnesota DuluthDept. of Biology and [email protected]

CP12

Modeling Population Spread in Heterogeneous En-vironments Using Integrodifference Equations

An integrodifference equation model for population dy-namics is presented. Dispersal is described by a PDE. Het-erogeneity consists of two types patches, one with a highgrowth rate and one with a low growth rate, arranged inpatch-work fashion. In this talk we address two question:(1) can a species persist and (2) at what rate will it spread?To address the first, we perform a linear stability analysis.To address the second, we compute a dispersion relationfor traveling wave solutions.

Thomas C. RobbinsDepartment of MathematicsUniversity of [email protected]

Mark LewisUniversity of Alberta, [email protected]

CP13

Blow-Up of Solutions of Degenerate QuasilinearParabolic Equations

Let p andm be real numbers such that p > m > 1, and q bea nonnegative real number. We study existence, uniquenessand blow-up of the solution of the following degeneratequasilinear parabolic problem

xqut = (um)xx + up in (0, 1) × (0, T ) ,

u (x, 0) = u0 (x) in [0, 1] , u (0, t) = 0 = u (1, t) for t ∈ (0, T ) ,

where u0 (x) is a smooth positive function and u0 (0) =u0 (1) = 0.

W. Y. ChanDepartment of MathematicsSoutheast Missouri State [email protected]

CP13

Singularities at Time Equal to Infinity in Equivari-ant Harmonic Map Flow

Many nonlinear parabolic equations in geometry and ap-plied mathematics (mean curvature flow, Ricci flow, har-monic map flow, the Yang-Mills flow, reaction diffusion

equations) have solutions which become singular either infinite or infinite time, meaning either that the evolvingobject (map, metric, surface, or function) becomes un-bounded, or that one of its derivatives becomes unbounded.The analysis of the asymptotic behaviour of a solution ofa nonlinear parabolic equation just before it becomes sin-gular is known to be a difficult problem. The main generalpoint of this talk is that this analysis is considerably easierin the case where the singularity occurs in infinite time.

Joost HulshofVU AmsterdamDepartment of [email protected]

Sigurd B. AngenentUniversity of WisconsinDepartment of [email protected]

CP13

Multi-Bump, Blowup, Self-Similar Solutions of theComplex Ginzburg-Landau Equation

In this talk we study, both asymptotically and numerically,multi-bump, blowup, self-similar solutions to the complexGinzburg-Landau equation in the limit of small dissipation.Furthermore, a proof of the existence of these multi-bump,blowup solutions will be given. Through the asymptoticanalysis, we determine the parameter range over whichthese solutions may exist. Most intriguingly, we determinea branch of solutions that are not perturbations of solutionsto the nonlinear Schrodinger equation, moreover, they arenot monotone but they are stable.

jf WilliamsLeiden UniversityDept of [email protected]

Chris BuddSchool of Mathematical SciencesUniversity of [email protected]

Vivi RottschaferLeiden UniversityDept of [email protected]

CP14

Direct Chaotic Flux Formula under General Time-Periodicity

Chaotic flux occurring across heteroclinics under nonsep-arable time-periodic (not necessarily harmonic) perturba-tions of area-preserving planar flows is examined. Thoughwell-understood phenomenologically, computable flux for-mulas have been lacking. By directly assessing the un-equal lobe areas that are transported via a turnstile mech-anism, such is obtained through a bi-infinite modulatedsummation of coefficients of the Melnikov function. Theseare themselves expressible using Fourier transforms. Com-putability under complicated perturbations is illustratedby example.

Sanjeeva BalasuriyaUniversity of SydneySchool of Mathematics & Statistics

82 DS05 Abstracts

[email protected]

CP14

Efficient Topological Chaos Embedded in theBlinking Vortex System

The particles mixings by blinking vortex system are stud-ied. It is well known that the chaotic advection occurs bythe system due to the homoclinic chaos. A braid is assignedfrom a well-controlled operation of the system, and suchbraids are classified into three types via Thurston-Nielsentheory. We propose a operation assigned a pseudo-Anosovbraid, which induces topological chaos, to realize efficientand global particles mixing in the whole space.

Takashi SakajoHokkaido UniversityDepartment of [email protected]

Eiko KinDepartment of mathematics, Kyoto [email protected]

CP14

Complex Basins in Weakly Dissipative DynamicalSystems

Chaotic scattering in open Hamiltonian systems underweak dissipation is not only of fundamental interest butalso important for problems of current concern such as theadvection and transport of inertial particles in fluid flows.Previous work using discrete maps demonstrated that non-hyperbolic chaotic scattering can be metamorphically in-fluenced by weak dissipation in that the algebraic decay ofscattering particles becomes exponential. Here we extendthe result to continuous-time Hamiltonian flows by usingthe Henon-Heiles system as a prototype model. We also gobeyond by investigating the basin structure of scatteringdynamics. We find, in the common case where multipledestinations exist for scattering trajectories, Wada basinboundaries are common and they appear to be structurallystable under weak dissipation even if the nonhyperbolicscattering dynamics is not.

Ying-Cheng LaiArizona State UniversityDepartment of [email protected]

Jesus M. SeoaneNonlinear Dynamics and Chaos Group.Universidad Rey Juan [email protected]

Miguel A.F. SanjuanNonlinear dynamics and chaos groupUniversidad Rey Juan [email protected]

CP15

Three-Wave Coupling in the Tokamak Turbulence

We use the Hasegawa-Mima equation to study the dynamicchange of the spatial Fourier spectrum of tokamak plasmaturbulence. The energy flow among the three coupled dom-inant spatial modes depends on the low energy injection ofthe stationary tokamak discharges. For a range of energy

injection, we find numerically recurrent oscillations of themode amplitudes that corresponds to chaotic or periodicattractors. A bifurcation diagram is presented as a func-tion of the energy injection.

Antonio BatistaDepartment of MathematicsState University of Ponta [email protected]

Sergio LopesDepartment of PhysicsFederal University of [email protected]


Ricardo VianaPhysics DepartmentFederal University of [email protected]

CP15

Frequency Locking in Extended Systems: the Im-pact of a Turing Mode

A Turing mode in an extended periodically forced oscilla-tory system can change the classical resonance boundariesof a single forced oscillator. Using the normal form equa-tion for forced oscillations, we identify a Hopf-Turing bi-furcation point around which we perform a weak nonlinearanalysis. We show that resonant standing waves can ex-ist outside the 2:1 resonance region of uniform oscillations,and non-resonant mixed-mode oscillations may prevail in-side the resonance region.

Arik YochelisDepartment of Chemical EngineeringTechnion, Haifa, [email protected]

Christian ElphickCentro de Fisica No Lineal y Sistemas Complejos deSantiagoSantiago, Chilenone

Ehud MeronDepartment of Energy and Environmental PhysicsBIDR, Ben Gurion University, Sede Boker Campus 84990,[email protected]

Aric HagbergLos Alamos National [email protected]

CP16

Dynamics of a Penning-Malmberg Trap

A grid-free multipole treecode method is used to explorecharged particle dynamics in a Penning-Malmberg trap.The system is formally equivalent to the 2D incompress-sible Euler equations of fluid dynamics. The interactionof the self-induced particle electric field with the confining

DS05 Abstracts 83

magnetic field leads to complex dynamics in this system.

Andrew J. Christlieb, Ronert KrasnyUniversity of MichiganDepartment of [email protected], [email protected]

CP16

Variational Averaging and the Guiding CenterEquations for Charged Particle Motion in a Mag-netic Field

We derive governing equations for the average motion ofcharged particles in a magnetic field. To this end, we ap-ply a novel procedure to average the variational principle.The resulting equations are equivalent to the guiding cen-ter equations for charged particle motion; this marks aninstance where averaging and variational principles com-mute. Finally, we compare our procedure with othersfor recovering averaged dynamics from multiscale systems,including Whitham averaging and coarse analysis (I. G.Kevrekidis).

Jerrold E. MarsdenCalifornia Inst of TechnologyDept of Control/Dynamical [email protected]

Jimmy FungCalifornia Institute of [email protected]

Harish S. BhatCalifornia Institute of TechnologyControl and Dynamical [email protected]

CP17

Solitons and Electric Conduction in Dissipative 1DLattices

The Toda lattice is known to possess soliton (cnoidal wave)solutions. With appropriate forcing these solitons may bemaintained even in the presence of dissipation, and if cou-pled to electrons via a Coulomb type interaction, can drivea “superconducting” current. Solitons have in fact beensuggested as a possible mechanism for high temperature su-perconductivity. We investigate these solitons and soliton-electron bound states in a simple 1D lattice, describingtheir origin and subsequent bifurcations for different typesof interaction potentials and forcing functions.

Jeff PorterUniversidad Complutense [email protected]

CP17

Dynamics of a Toda Lattice with Weak Viscosity

I will consider a one-dimensional system which consists ofnonlinear springs with a Toda potential and linear dash-pots. In this system, the dash-pots work as the sourceof weak viscosity. Thus, in short time scale, the systembehaves like an usual Toda lattice, while in long time scale,it behaves like a viscous fluid. I will present the equationsof motion, and discuss the dynamics of the system.

Toshio Yoshikawa

Academia SinicaInstitute of [email protected]

CP18

Bifurcations in a Highway Traffic Model withDrivers’ Reaction-Time Delay

A car-following model of highway traffic involving thereaction-time delay of drivers is investigated. Bifurca-tions of the corresponding system of delay differential equa-tions are studied analytically and by numerical continua-tion techniques. Regions in the parameter space are deter-mined, where the equilibrium coexists with periodic solu-tions corresponding to single or multiple traffic jams. Thelow-dimensional dynamics of amalgamation and dispersionof traffic jams are also investigated.

Gabor OroszUniversity of [email protected]


Bernd KrauskopfUniversity of BristolDept of Eng [email protected]

CP18

Periodic Gaits of a Quasi-Passive Dynamic WalkingRobot with Two Legs and Feet

We consider a simple walking robot which has two legsand feet and can kick the ground, and show that it canwalk on the horizontal plane and even ascend gentle slopes.Moreover, its energy consumption is very low, comparedwith the traditional approaches. To this end, we reducethe problem of obtaining such periodic gaits to a nonlin-ear boundary value problem (BVP) for ODEs, and use acomputer software called AUTO to numerically analyze thenonlinear BVP.

Yasuhisa HasegawaUniversity of TsukubaGraduate School of Systems and Information [email protected]


CP19

A Flow for Data Clustering

Data clustering is an increasingly important problem forengineering, biological, sociological and other sciences. Wepresent a dynamical system which evolves on an adjointorbit of the Lie group SO(N) and performs data clustering.Our objective is not to provide the most efficient algorithm,but to give an original approach and shed some new lighton this long known problem.

Mohamed Ali BelabbasHarvard University

84 DS05 Abstracts

[email protected]

CP19

Nonlinear Time Series Prediction Using LocalModeling: a Comparison of Different Approaches

Predicting the future evolution of dynamical systems is amajor goal in many areas of science. Often the underly-ing dynamical equations are unknown and only a single-channel time series is available. Prediction methodologiesbased on embedding and attractor reconstruction using lo-cal statistical models constructed from the data are well-established. A variety of approaches for building local mod-els from data has been proposed: local polynomial modelsbased on nearest neighbors, radial basis function models,cluster-weighted modeling, adaptive local polynomial mod-els. The present study compares/contrasts all these differ-ent approaches on real observational data from an elec-tronic circuit. Both best guess and probabilistic predictionis considered.

Leonard SmithOCIAMUniversity of [email protected]


CP19

Towards a Non-Linear Trading Strategy for Finan-cial Time Series

A new trading strategy is proposed. The technique usesthe state space volume evolution and its rate of change asindicators. This methodology has been tested off-line usingeighteen high-frequency foreign exchange time series withand without transaction costs. In our analysis an optimummean value of 25% gain may be obtained in those serieswithout transaction costs and an optimum mean value of11% gain assuming 0.2% of costs in each transaction

Fernanda StrozziCarlo Cataneo University (LIUC)Quantitative Methods [email protected]

Jose Manuel ZaldivarEuropean Commission, Joint Research Centre, [email protected]

CP21

Simulation-Based Methodology and Tools for Anal-ysis of Electric Power Networks

We present a computer-based methodology for studyinglarge-scale electric power networks. An efficient and ver-satile matlab code forms the foundation of the assessmentprocedures providing dynamic simulation and analysis ca-pabilities. The methodology integrates a variety of tech-niques based on geometric nonlinear control theory, pa-rameter estimation methods, and other dynamical systemanalysis tools. These strategies were developed to addressmodel development and validation, vulnerability assess-ment and mitigation, long-term planning, and real-time

system evaluation.

Kevin WedewardNew Mexico TechElectrical Engineering [email protected]

Steven [email protected]

Steve SchafferNew Mexico TechMathematics [email protected]

Ernie BaranyNew Mexico State UniversityDepartment of Mathematical [email protected]

CP21

Effects of Degree Correlation on Network Synchro-nizability

The syncronization of networks of coupled oscillators hasdrawn recently a lot of interest. It has been shown that thenetwork structure seems to be able to affect the stabilityof the sinchronized state. The scale free nature character-istic of real world networks has been shown to be able toworsen the synchronizability, as an effect of the strong het-erogeneity of the connectivity. In order to explain this phe-nomenon, the spectrum of the Laplacian of the graph hasbeen widely investigated and a relationship has been foundbetween the largest-smallest eigenvalue ratio and synchro-nizability. Lately, it has been discovered that many realnetworks are characterized by degree correlation, that isthe tendency for vertices to be connected to other verticeswith similar degrees. A negative correlation (or disassor-tativity) is typical of technological networks such as theInternet and the World Wide Web, in which high degreenodes are more likely to be connected to low degree ones.Here we will study the relationship between degree corre-lation and network sinchronizability and we will show thatnegative correlation can have positive effects on networksinchronizability. We will discuss the effects of the networkassortativity properties on the synchronization dynamics.

Francesco SorrentinoUniversita’ di Napoli Federico [email protected]

Mario Di BernardoUniversity of BristolDept of Engineering [email protected]

CP21

The Fermi-Dirac Distribution and Traffic in Com-plex Networks

We propose an idealized model for traffic in a network, inwhich many particles move randomly from node to node,following the network’s links, and it is assumed that atmost one particle can occupy any given node. We showthat the particles behave like free fermions, and their statis-tical properties are given by the corresponding Fermi-Diracdistribution. We use this to obtain analytical expressions

DS05 Abstracts 85

for dynamical quantities of interest.

Alessandro MouraInstitute of PhysicsUniversidade de Sao [email protected]

CP22

Reentrant Hexagon Patterns in Non-BoussinesqConvection

The classical scenario for non-Boussinesq convection con-sists of hexagonal patterns at onset and a transition toroll patterns somewhat above onset. We investigate thestrongly nonlinear regime numerically for various fluids andfind quite generally in the roll regime a second bifurca-tion, which restabilizes the hexagons. For stronger non-Boussinesq effects the two transitions merge, eliminatingthe instability of hexagons to rolls altogether. This sce-nario is captured qualitatively in an extension of the usualamplitude equations.

Hermann RieckeApplied MathematicsNorthwestern [email protected]

Santiago MadrugaNorthwestern UniversityApplied [email protected]

Werner PeschU. BayreuthTheoretische [email protected]

CP22

Spatio-Temporal Chaos of Hexagon Patterns in Ro-tating Non-Boussinesq Convection

In rotating systems the transition from hexagonal convec-tion patterns to rolls is replaced by a Hopf-bifurcation tooscillating hexagons. For strongly non-Boussinesq convec-tion in water we find that the Hopf-bifurcation can be-come subcritical. The ensuing bistability between oscil-lating and steady hexagons leads to spatio-temporal chaoscharacterized by chaotically evolving domains of oscillatinghexagons amidst steady hexagons. Since these hexagonalpatterns are defect-free, the spatio-temporally chaotic dy-namics might be captured by a quintic complex Ginzburg-Landau equation.


Santiago MadrugaNorthwestern UniversityApplied [email protected]

Werner PeschU. BayreuthTheoretische [email protected]

CP22

Oscillatory Pattern Formation with a ConservedQuantity

The influence of a conserved quantity on an oscillatorypattern-forming instability is examined in one space dimen-sion. Amplitude equations are derived which are generic forsystems with a pseudoscalar conserved quantity (e.g., ro-tating convection, magnetoconvection) but are also appli-cable to systems with a scalar conserved quantity. The sta-bility properties of travelling and standing waves are anal-ysed and new long-wavelength instabilities are reported.These instabilities give rise to modulated waves and highlylocalised pulse-like solutions.

David M. WinterbottomUniversity of [email protected]

CP23

Bifurcations on the Sphere with Inhomogeneity

We study the steady-state pattern formation problem withspherical symmetry and inhomogeneity. Group theory re-veals the most general normal form for a system in onerepresentation of O(3), subject to an explicit symmetrybreaking that belongs to another representation. For themorphogenesis problem we find one viable scenario for theformation of the antero-posterior and dorso-ventral axes:that with both the system and the inhomogeneity belong-ing to the l = 1 representation.

Timothy K. CallahanArizona State UniversityDepartment of Mathematics and [email protected]

CP23

Hopf Bifurcation from Rotating Waves on theSphere

Spiral waves are spatio-temporal patterns that have beenobserved in numerous physical situations, ranging fromBelousov-Zhabotinsky chemical reactions to cardiac tissue.The global geometry of the heart is closer to a sphere thanto a plane. Therefore, it would be important to study spiralwave dynamics in a context where the symmetries are thoseof the sphere instead of the plane. Also, rigidly rotatingspiral waves are rotating waves. In this talk, we considerone-parameter dependent reaction-diffusion systems on thesphere of radius r, which are equivariant under the groupSO(3) of all rigid rotations:

∂u

∂t(t, x) = D∆u(t, x) + F (u(t, x), λ) on rS2, (2)

u = (u1, u2, . . . , uN ) with N ≥ 1, D is the diffusionmatrix and F = (F1, F2, . . . , FN ) is a sufficiently smoothfunction.

For λ = 0 we consider a relative equilibrium SO(3)u0 withtrivial isotropy Σu0 = I3 consisting of rotating waves. ByHopf bifurcation from rotating waves, we generically getmodulated rotating waves. We show that there exists adecomposition of these modulated rotating waves in twoparts: a primary frequency vector part, eX(λ)t and a 2π

|ωλ| -periodic part, B(t, λ). We present a way of constructingX(λ) and B(t, λ) using the BCH formula in so(3) and thesolution Z(t, λ) ∈ so(3) to the initial value problem on

86 DS05 Abstracts

some interval [0, 2πn|ωλ| ], with the integer n > 0 indepen-

dent of λ:

−→Z =

[I3 + 1

2Z +

(1

|Z|2 − cos|Z|2

2 sin|Z|2

|Z|

)Z2

]−−−−−→XG(t, λ),

Z(0) = O3.(3)

This decomposition allows us to show that the quasi-periodic meandering of the modulated rotating waves ob-tained by Hopf bifurcation is possible and, in fact, thatthere are three types of motions for the tips of these modu-lated rotating waves. These types of motions are visualizedusing Maple.

In the case of a resonant Hopf bifurcation from a rotat-ing wave in two-parameter dependent SO(3)-equivariantreaction-diffusion systems on the sphere, we obtain abranch of modulated rotating waves with primary fre-quency vectors orthogonal to the frequency vector of therotating wave undergoing Hopf bifurcation.

Adela ComaniciUniversity of [email protected]

CP23

Tissue Dynamics - Modeling and Experiment

Early detection of cancer is critical to improved success intreatment. Much progress has been made in understandingthe molecular and genetic underpinnings of cancer. Less isknown about how cells, as collective populations, progresstowards a malignant state. This talk will review some ofthe models that treat tissues as dynamical systems, with afocus on epithelial tissues such as the lining of the mouth.Our goal is to design experiments that explore dynamicaland pattern forming aspects of tissues as they continuallyremodel themselves, both in healthy and diseased states.Progess in our experimental characterization will be de-scribed.

Randall TaggUniversity of Colorado at DenverDepartment of [email protected]

CP24

Singularly Perturbed Cluster Growth in Aggrega-tion Kinetics

In the classical Becker–Doring model of aggregation ki-netics, each cluster assumes “critical size” relative to itssurface monomer concentration. This matching betweencluster size and surface concentration seems energeticallyunstable. However, careful analysis reveals the stability ofthe matching. Cluster growth is described by a singularlyperturbed diffusion BVP. Using different time-scales, thematching can be shown to be stable. This analysis is onestep in a larger ongoing singular perturbation analysis ofaggregation kinetics.

Yossi FarjounUC [email protected]

CP24

Diffusion-Limited Reaction in 1 Dimension: Exact

Results and Numerics

We study the dynamics of diffusing particles in one spacedimension with annihilation on collision and random nu-cleation (creation of particles). A new method of analysispermits exact calculation of the density of particles. Withpaired nucleation at sufficiently small initial separation thenucleation rate is proportional to the square of the steadystate density. With unpaired nucleation, the nucleationrate is proportional to the cube of the steady state density.

Grant LytheDepartment of Applied MathsUniversity of [email protected]

CP25

Perturbing Flows for Optimum Chaotic Flux

Motivated by optimizing mixing in microfluidic devices, thenature of the time-periodic perturbation subject to a C0

bound which would generate the maximum chaotic fluxacross a heteroclinic separatrix is investigated. Though anill-posed optimization problem, it is shown that choosingthe perturbation as close as possible to a certain (unphysi-cal) flow generates the best mixing. An expression for thismaximum chaotic flux approachable at each frequency ofperturbation is obtained and analyzed.

Sanjeeva BalasuriyaUniversity of SydneySchool of Mathematics & [email protected]

CP25

Hamiltonian Theory of Mixing in a Microfluid De-vice

Mixing of fluids in microchannels cannot rely on turbu-lence since the flow takes place at extremly low Reynoldsnumbers. Various active and passive devices have beendeveloped to induce mixing in microfluid flow devices. Wedescribe here a model of an active mixer where a transverseperiodic flow interacts with the main flow in R2. We de-velop a Hamiltonian description of the dynamics. We provenonintegrability of the system, describe adiabatic invari-ants, show the existence of KAM-tori, and examine somespecific solutions.

Mikhail V. DeryabinFaculty of Mathematics and MechanicsMoscow State [email protected]

Poul G. HjorthTechnical Univ of DenmarkDepartment of [email protected]

CP25

Describing Homoclinic Tangles Through Homo-topic Lobe Dynamics

Homoclinic tangles are basic geometric structures that or-ganize the dynamics of invertible maps on the plane. Thetopology of such tangles provides an approach to under-standing the qualitative dynamics of transport and escapephenomena. Since the original introduction of tangles byPoincare, a number of researchers have characterized spe-

DS05 Abstracts 87

cific classes of tangles. However, there seems to be a lackof general techniques applicable to the wide variety of tan-gles realized by maps. To address this issue, we present anew symbolic technique for describing tangles. This tech-nique, called homotopic lobe dynamics, allows a homoclinictangle to be described up to arbitrarily long times, or al-ternatively, to arbitrarily small scales.

John DelosCollege of William and [email protected]

Kevin A. MitchellUniv of California, MercedSchool of Natural [email protected]

CP26

Stalking: Aggressive Shadowing of a Noisy Trajec-tory

Shadowing theory addresses the question of how trajecto-ries of a low-resolution model L compare with solutions ofa high-resolution model H. For typical H, no trajectory ofL remains close to an H trajectory for all time. There-fore, we examine perturbed trajectories of L. We find suchpseudo-trajectories of L can remain close to H (for a rea-sonable definition of ’close’) even when the perturbationsare much smaller than the difference between L and H.Pseudo-trajectories of a low-resolution model L shadow anH trajectory for orders of magnitude longer than true tra-jectories of L. We discuss applications of stalking trajecto-ries to weather forecasting.

James A. YorkeUniversity of MarylandDepartments of Math and [email protected]

Chris M. DanforthApplied Mathematics and Scientific ComputationUniversity of [email protected]

CP26

Convex Error Growth Patterns in a High Dimen-sional Chaotic Dynamical System for Weather Pre-diction

The error growth, that is the growth of the distance be-tween two almost identical solutions of a high dimensionalchaotic dynamical system, is studied and compared tothose with lower dimensions. Typically, the exponentialgrowth rate plateaus and decreases concavely in amplitude-wise asymptotic to a log-linear line. In contrast, we findthat in a higher dimensional system, the error growth ratedecreases as if a convex function in amplitude-wise andclose to two log-linear lines.

John HarlimDepartment of MathematicsUniversity of [email protected]

Brian R. Hunt, James YorkeUniversity of [email protected], [email protected]

Michael Oczkowski

Department of Physics and AstronomyFrancis Marion University, Florence SC [email protected]

Eugenia KalnayDepartment of MeteorologyUniversity of [email protected]

CP26

A Filtration Algorithm for Computing LyapunovExponents

Recently, Ott and Yorke described the construction of “en-veloping manifolds” to answer the following question: IfA is an arbitrary compact set in Rn, what is the smallestinteger d such that, given x ∈ A, there is a compact C1

manifold of dimension d that contains all points of A thatlie in some neighborhood of x? In this talk, we describehow to apply this machinery to the computation of Lya-punov exponents from chaotic time series. In particular,we discuss how spurious exponents can be identified andeliminated.

Eric J. KostelichArizona State UniversityDepartment of [email protected]

CP27

Dispersive States in Propagative Systems with Pe-riodic Structure

I will analyze the weakly nonlinear dynamics of a propaga-tive spatially extended system with spatial periodic struc-ture. Light propagation in fiber Bragg gratings and Bose-Einstein condensates in optical lattices are two examplesof systems with spatial periodic structure that have re-cently received very much attention. The nonlinear reso-nant dynamics of these systems is dominated by two maineffects: the counter-propagating transport (that is coupledthrough the periodic structure) and the dispersion, whichhas been systematically neglected in previous studies basedon the nonlinear coupled mode equations description. Iwill show that dispersion gives rise to new instabilities,complex spatio-temporal patterns and new localized struc-tures, which cannot be detected if dispersion effects are notconsidered. I will also present some numerical simulationsto illustrate these essentially dispersive states.

Carlos MartelUniversidad Politecnica de [email protected]

CP27

Convection Induced Spatial Nonlinear Phase Mod-ulations in Optical Systems

We show that the combined action of diffraction andconvection (walk-off) leads to a nonlinear phase modu-lation in degenerate optical parametric oscillators. Nearthreshold, this mechanism is described analytically by aGinzburg-Landau amplitude equation where the nonlinearself-coupling coefficient depends on the convection term.This enable us to characterize the induced asymmetry inthe generated travelling waves. The predictions are in ex-cellent agreement with the solutions of the governing equa-

88 DS05 Abstracts

tions.

Roberta ZambriniDepartment of PhysicsUniversity of [email protected]

Celine DurniakPhLAM - Universite de Lille (France)[email protected]

Maxi San MiguelInstituto Mediterr’aneo de Estudios Avanzados IMEDEA(CSIC-UIB)[email protected]

Majid Taki

PhLAM - Universite de Lille (France)[email protected]

CP28

3D Modeling and Surface Oscillation of a LevitatedDroplet

The effect of the surface oscillations for a levitated (EML)droplet is considered in this study. The droplet is mod-elled as a three dimensional system with lumped massesand elastic springs. The expression for the spring elas-tic constants in a global stiffness matrix and the surfaceoscillations modes are presented. It is shown that adjust-ments to the magnetic field, enable the levitated specimento approach to a spherical shape and improve the abilityto control the stability. Computations were performed fordroplets of aluminum and copper.

Mihai Dupac, Davig Beale, Ruel OverfeltAuburn [email protected], [email protected],[email protected]

CP28

Geometrical Formulation of Multilevel MesoscopicDynamics and Thermodynamics

To interpret multilevel experimental observations of com-plex fluids we need a multilevel (multiscale) models. Ageneral framework for such models is provided by contactgeometry. The mesoscopic time evolution is the time evolu-tion preserving the contact structure of mesoscopic thermo-dynamics. Particular realizations of such abstract dynam-ical system include, among others, classical and extendednonequilibrium thermodynamics and the Boltzmann ki-netic theory. In this talk the abstract mesoscopic dynamicsis illustrated in the context of multilevel modelling of tur-bulent flows.

Miroslav GrmelaEcole Polytechnique de [email protected]

CP28

A Concurrent Multi-Scale and Multi-Physics Mod-eling of Biological Systems

A major challenge in the modeling of biological systems isthe accurate description of physical, chemical, and biolog-ical phenomena over a wide range of spatial and tempo-

ral scales. In practice, various hierarchical modeling tech-niques have been explored. However, the concurrent cou-pling which is more relevant and ubiquitous in biologicalsystems has not yet been fully understood. In this talk,a mathematical framework of concurrent multi-scale andmulti-physics modeling of biological systems is laid out. Inparticular, two specific aspects of this modeling strategywill be elaborated. Firstly, the validity of the newly de-veloped immersed continuum method (ICM) will be exam-ined via a meshless finite element solid model coupled withthe surround fluid. This new modeling method enables aunified treatment of flexible structures or solids moving inaqueous environment and the direct coupling with massand heat transfer equations. Secondly, a finite tempera-ture bridging scale approach is presented to link macro-scopic computational mechanics with localized refinementsat micro-or nano-scale using molecular dynamics. Finally,future work spanning the quantum and continuum lengthscales will be proposed.

Sheldon WangDepartment of Mathematical SciencesNew Jersey Institute of [email protected]

CP29

Structural Stability of Boundary Equilibria

In recent years, much research effort in applied scienceand engineering has focussed on dynamical systems whichare nonsmooth or discontinuous. This lecture is concernedwith the structural stability of boundary equilibria in non-smooth dynamical systems. Namely, we study the struc-tural stability under parameter variations of equilibria ly-ing on discontinuity boundaries in phase space dividing re-gions where the system under investigation is smooth. Weshow that it is possible to give a set of conditions to accountfor the possible dynamical scenarios that can be observed.In particular, we are interested in isolating the branchesof solutions originating from a boundary equilibrium pointunder parameter variations.

Mario Di BernardoUniversity of BristolDept of Engineering [email protected]

CP29

Dynamics of Stochastic Nonsmooth Systems

Many power electronics systems are nonsmooth (egDC/DC boost and buck converters). These systems canundergo changes in behaviour under parameter variationthat are not present in smooth systems called grazing bi-furcations. We highlight the dramatic effect that noise canhave on these bifurcations (including advance or smooth-ing of the bifurcation) and conclude by showing excellentagreement between experimental results and a stochasticmodel for the DC/DC boost converter.

John HoganBristol Centre for Applied Nonlinear MathematicsDepartment of Engineering Mathematics, University [email protected]

CP29

Continuous and Discontinuous Grazing Bifurca-

DS05 Abstracts 89

tions in Quasi-Periodic Oscillations

Grazing bifurcations, occuring due to tangential contactbetween a system attractor and state-space discontinu-ities, may cause dramatic changes in system dynamics.Although its well known that grazing bifurcations of pe-riodic system attractors can be either continuous or dis-continuous, this work reveals that grazing bifurcations ofquasi-periodic attractors are never strictly discontinuous innature. Numerical results are presented for a 2D torus in4D state space, where continuous and apparently discon-tinuous bifurcations arise for various parameter values.

Phanikrishna ThotaVirginia Polytechnic Institute and State [email protected]

Harry DankowiczVirginia Polytechnic Institute and State UniversityEngineering Science and [email protected]

CP30

Dynamics of Conflicting Decision-Making Groups

We present a model of the interaction of two rival leader-ship groups engaged in conflict. Each group is representedas a hierarchical social network and modeled by a set ofcoupled nonlinear differential equations which evolve eachmembers policy position with respect to individual prefer-ence, group influence, and the policy of the opposing group.The model is based on social and cognitive psychology the-ories of attitude change, group dynamics, and conflict andrepresents an alternative approach to the two-level gameproblem which seeks to account for the interaction betweendomestic and foreign policies in negotiations.

Michael GabbayInformation Systems [email protected]

CP30

Scaling, Renormalization, and Universality inCombinatorial Games: the Geometry of Chomp

Combinatorial games, which include chess, go, checkers,Nim, Chomp, and dots-and-boxes, have both captivatedand challenged mathematicians, computer scientists, andplayers alike. Using Chomp as an archetype, we report ona new geometrical approach which unveils some unexpectedparallels between combinatorial games and key ideas fromphysics and dynamical systems, most notably notions ofscaling, renormalization, universality, and chaotic attrac-tors.

Adam S. LandsbergClaremont McKenna College, Pitzer College, [email protected]

Eric FriedmanORIECornell [email protected]

CP30

Control of Chaotic Transients: Yorke’s Game of

Survival

We consider the tent map as the prototype of a chaoticsystem with escapes. We show analytically that a small,bounded, but carefully chosen perturbation added to thesystem can trap forever an orbit close to the chaotic saddle,even in presence of noise of larger, although bounded, am-plitude. This problem is focused as a two-person, mathe-matical game between two players called “the protagonist”and “the adversary”. The protagonist’s goal is to survive.He can lose but cannot win; the best he can do is sur-vive to play another round, struggling ad infinitum. Inabsence of actions by either player, the dynamics diverge,leaving a relatively safe region, and we say the protagonistloses. What makes survival difficult is that the adversaryis allowed stronger “actions” than the protagonist. Whatmakes survival possible is (i) the background dynamics (thetent map here) are chaotic; and (ii) the protagonist knowsthe action of the adversary in choosing his response and ispermitted to choose the initial point x0 of the game. Weuse the “slope 3” tent map in an example of this problem.We show that it is possible for the protagonist to survive.

Jacobo AguirreUniversidad Rey Juan Carlos, [email protected]

Miguel A. SanjuanNonlinear Dynamics and Chaos GroupUniversidad Rey Juan [email protected]

Francesco d’OvidioIMEDEA (CSIC-UIB)Universitat Illes Balears, E-07071 Palma de Mallorca,[email protected]

CP31

Bifurcations in Integrable and Near-IntegrableHamiltonian Systems

Topological properties of bifurcations in low-dimensionalHamiltonian systems are investigated. We give a listof generic possible singularities appearing in the energy-momentum bifurcation diagrams of a certain class of inte-grable and near-integrable systems and provide the classi-fication of the corresponding bifurcations. Also, the topo-logical description of energy surfaces at bifurcation statesis given.

Milena RadnovicFaculty of Mathematics and Computer ScienceThe Weizmann Institute of Science, [email protected]

Vered Rom-KedarWeizmann Instituteof [email protected]

CP31

Hierarchy of Bifurcations in the Truncated andForced NLS Model.

We analyze the truncated forced non-linear Schrodinger(NLS) model using a novel framework in which a hierarchyof bifurcations is constructed. Consequently, we provideinsights regarding its global structure and the type of in-

90 DS05 Abstracts

stabilities which appear in it. In particular, we show that aparabolic resonance mechanism of instability arises in therelevant parameter regime of this model. The analogousbehavior in the NLS pde is discussed.

Eli ShlizermanThe Weizmann Institute of ScienceDepartment of Computer Science and [email protected]


CP31

Data Assimilation and Hamiltonian Systems

Weather systems can, in part, be described by Hamilto-nian mechanics. Numerical weather prediction models usedata assimilation to estimate the optimal initial conditionsby combining observations with dynamical constraints us-ing control theory. We investigate whether the geomet-ric features inherent in Hamiltonian systems can be ex-ploited when implementing data assimilation. We use therestricted three body problem as a test bed for studyingconstraints in control problems with a Hamiltonian struc-ture.

Laura R. WatkinsonUniversity of [email protected]

Mike CullenMet [email protected]

Amos LawlessUniversity of [email protected]

Ian RoulstoneUniversity of [email protected]

Nancy K. NicholsUniversity of ReadingDepartment of [email protected]

CP32

Proof of Communication and Synchronization inDisconnected But Fast Moving-Neighborhood Net-works

We model a time-varying ad-hoc network of moving agentswhich move according to a dynamical system. Instanta-neously, the network is disconnected, but nonetheless themoving averaged graph Laplacian allows for communica-tion in the form of synchronization to occur. In this sense,we have generalized the master-stability function formal-ism to time varying networks. We prove that fast enoughvarying connections relative to information flowing acrossthe network compensates competing time scales betweenagents and oscillation.

Erik Bollt

Clarkson [email protected]

Joe SkufcaDept. MathematicsUS Naval [email protected]

Dan StillwellVirginia [email protected]

CP32

Measuring Network Structure and Its Effect onNetwork Dynamics.

Disease spread over contact networks is one example ofdynamics on networks. We use simple models of diseasespread (SI or SIR) to investigate how the spread dependson characteristics of the network structure at the site ofinitial infection. The notion of a backbone of the networkis discussed and attempts made to identify it. Effects ofinoculation of this backbone on the spread suggests a po-tentially fruitful prevention strategy.

Daniel A. SchultColgate UniversityDept of [email protected]

CP32

Dynamics of Epidemics on Correlated Networks

A major and unexplained difference between social net-works and biological/technological networks is the positiveassortativity (degree-degree correlation) of the former andthe negative assortativity of the latter. We will discussthe structural constraints between clustering and degree-degree correlation and the resulting effects on dynamics ofepidemics and reaction-diffusion processes on networks.

Aric Hagberg, Pieter SwartLos Alamos National [email protected], [email protected]

CP33

Localised Convection Cells in the Presence of aVertical Magnetic Field

Thermal convection in a horizontal fluid layer heated uni-formly from below usually produces a spatially periodic ar-ray of convection cells of roughly equal amplitudes. In thepresence of a large scale vertical magnetic field, convectionmay instead occur in vigorous isolated cells, separated byregions of strong magnetic field. Convection in this formmay be responsible for the appearance of bright umbraldots in the centres of sunspots. This talk will discuss amodel for two-dimensional localised solutions of this kindin which the amplitude of convection is not assumed to besmall. The model reflects the physics of the interaction be-tween field and flow, and elucidates the approximate regionof parameter space over which these solutions exist.

Jonathan DawesDAMTP, University of [email protected]

DS05 Abstracts 91

CP33

Boundary Effects and the Onset of Taylor Vortices

The onset of spatially periodic vortex states in the Taylor–Couette flow between rotating cylinders occurs at the valueof Reynolds number predicted by local bifurcation theory.However, the symmetry breaking induced by the top andbottom plates means that the true situation should be adisconnected pitchfork. This leads to an apparent contra-diction: why should Taylor vortices set in so sharply at theReynolds number predicted by the symmetric theory, givenlarge symmetry-breaking effects caused by the boundaryconditions? We offer a generic explanation, based on aSwift–Hohenberg pattern formation model that shares thesame qualitative features as the Taylor–Couette flow.


Alastair M. RucklidgeDepartment of Applied MathematicsUniversity of [email protected]

CP33

Thermocapillary Controlled Rupture of Thin Liq-uid Sheets

We study the rupture process of a fluid sheet induced by aspatially distributed initial temperature profile. For vari-cose disturbances, analysis of a long-wave approximation(yielding a coupled system of nonlinear evolution equa-tions) shows that film rupture can be controlled by carefulselection of the Marangoni number. We extend these ideasto sinuous disturbances, discuss the implications of the re-sults to cylindrical jets and also comment on importantindustrial applications of the work.

Burt S. TilleyFranklin W. Olin College of [email protected]

Mark BowenSchool of Mathematical SciencesUniversity of [email protected]

CP34

Spontaneous Activity in the Visual Cortex

I will present experimental results (Grinvald et al.) illus-trating the spontaneous background activity in the visualcortex. Then I will present our model for the visual cortex,and briefly discuss the computational problems associatedwith this model, and our numerical method. I will use ourmodel to explain this spontaneous bacground activity, andpresent predictions of the model. If I have time, I’ll goover other scale-up techniques, and the numerical methodsassociated with them.

Aaditya RanganCourant Institute of Mathematical SciencesNew York [email protected]

CP34

Stochastic Resonance in Vision: Models and Data

Stochastic resonance (SR) is a phenomenon through whichsmall amounts of additive noise enhance the performance ofa signal processing system. This paradoxical enhancementis to be expected whenever the system has nonlinearities.Recently the phenomenon has found exciting applicationsin sensory physiology, where nonlinearities in biologicalsensory transducers make SR ubiquitous. We report on re-cent experimental and modelling studies that demonstratethe extent to which the human visual system is adapted touse SR.

Carole L. Tham, Mark MuldoonThe University of [email protected], [email protected]

CP34

Waves in Turtle Visual Cortex Have Three Com-ponents

Visual stimuli evoke propagating waves in turtle visual cor-tex . A large-scale model was used to characterize theactivity of cortical neurons in response to simulated lightflashes. Pyramidal cell firing shows three components: aninitial depolarization, a primary propagating wave and asecondary wave. The transitions from the initial depo-larization to primary waves and from primary to the sec-ondary waves are controlled by separate populations of in-terneurons. (Supported by the NSF CRCNS program)

Philip UlinskiThe University of Chicago, ChicagoDepartment of Organismal Biology and [email protected]

CP35

Multiple Equilibria in Biochemical Reaction Net-works and the Jacobian Criterion

The Jacobian criterion examines the capacity for multi-ple equilibria of high dimensional dynamical systems givenby some classes of functions, for example polynomial func-tions. In particular, we generate methods that allow us todecide whether a given biochemical reaction network hasthe capacity for multiple equilibria, and to identify sub-networks that give rise to multiple equilibria. Also, wediscuss implications for the interpretation of experimentsin cell biology. This is joint work with Martin Feinberg.

Gheorghe CraciunMathematical Biosciences Institute, Ohio State [email protected]

CP35

Constrained Hybrid Monte-Carlo and Free EnergyCalculation

We address the problem of computing the potential ofmean force (free energy) for a prescribed set of curvilin-ear coordinates in a molecular system. The approach givesconcise geometrical insight into the different contributionsto the mean force and we provide a hybrid Monte-Carlobased algorithm to compute the mean force using con-strained simulations.

Carsten HartmannFree University Berlin

92 DS05 Abstracts

Institute of [email protected]

CP35

Pattern Formation and Surface Design for Het-erogenous Reactions on Structured Surfaces

Reaction diffusion models for heterogenous catalytic re-actions on structured surfaces are considered. Space-dependent coefficients represent different types of materialsand the spatiotemporal patterns of the system depend onthis surface structure. The surface design is optimized, i.e.a certain amount of an expensive catalyst material is dis-tributed on a cheaper carrier material such that the outputof the chemical reaction is maximized while considering thedynamical properties of the system.

Jens StarkeUniversity of HeidelbergInstitute of Applied [email protected]

CP36

Low Order Modeling, Dynamics, and Control ofFlow Separation

We present efforts to derive low order models, analyze dy-namics, and synthesize controllers for a class of fluid sepa-ration problems dominated by coherent vortex structures.Due to POD’s inability to resolve patterns not present inthe flow when the basis function were derived, we pursuea different, Lagrangian approach. The resulting systemsof nonlinear ODEs captures key features of the flows. Thestructure of the ODEs are unusual, but amenable to anal-ysis and control synthesis.

Brianno CollerNorthern Illinois [email protected]

CP36

Adiabatic Invariance and Geometric Angle in Two-Dimensional Fluids

We examine the response of two-dimensional, incompress-ible and inviscid fluids to slow deformations of the bound-ary of their domain. Two issues are addressed using per-turbation techniques: (i) the determination of the (leading-order) Eulerian flow fields, which are shown to depend onlyon the shape of the fluid domain at any given time, and(ii) the determination of Lagrangian particle trajectories,which depend on the successive shapes taken by the do-main. We emphasise the similarity between this problemand some familiar finite-dimensional analogues, such as thependulum with varying length, and in particular the con-nection between (ii) and the Hannay–Berry (geometric) an-gles. We present explicit results for nearly axisymmetricflows in slightly deformed discs.

Jacques VannesteSchool of MathematicsUniversity of [email protected]

Djoko WiroseotisnoDepartment of Mathematical SciencesUniversity of [email protected]

CP36

Elliptical Instability at Large Aspect Ratio

Elliptical instability is a universal mechanism by which pla-nar flows with elliptical streamlines become unstable tothree dimensional perturbations. The system is governedby two parameters µ and ε which are aspect ratios of thedomain. Previous work has largely focused on near ax-isymmetry cases (ε ≈ 0) using perturbation theory. Usingsymmetries, we develop a non-perturbative approach thatallows us to obtain exact description of the boundaries ofinstability wedges in the µ–ε plane. This analysis revealsa complex structure of the parameter plane, including abifurcation point at large aspect ratio where a seeminglyinfinite number of wedge boundaries intersect.

Norman LebovitzUniversity of ChicagoDepartment of [email protected]

Lay May YeapUniversity of ArizonaDepartment of [email protected]

CP37

Explicit Periodic Solutions in a Model of a RelayController with Delay and Forcing

We use a combination of numerical and analytical meth-ods to find and construct solutions of a model of relaycontrol, formulated as a piecewise-constant delay differ-ential equation (DDE). Numerical solutions of a relatedequation, where the discontinuities of the original DDE aresmoothed, are used to guide the construction of explicit so-lutions of the original DDE. Conversely, the constructionof explicit solutions provides starting data for numericalcontinuation of the smoothed equation.

David A. Barton, Eddie WilsonUniversity of Bristol, [email protected], [email protected]


CP37

Locating Periodic Orbits in High-Dimensional Sys-tems by Stabilising Transformations

An algorithm for detecting unstable periodic orbits(UPO’s) that combines a modified semi-implicit Eulermethod and a set of stabilising transformations has hadconsiderable success in low dimensional chaotic systems.Applying the same ideas to higher dimensional systems isnon-trivial due to the rapid increase in the number of trans-formations. We have proposed a much smaller set of trans-formations based on the stability properties of the UPO’sand successfully applied it to systems with dimension upto six. In this talk we explore the possibility of extend-ing this approach to high-dimensional flows representingPDE’s such as, for example, the Kuramoto-Sivashinski sys-tem.

Jonathan J. Crofts, Ruslan L. DavidchackUniversity of Leicester

DS05 Abstracts 93

[email protected], [email protected]

CP37

Approximating Limit Cycles of An AutonomousDelay Differential Equation

This talk presents an existence theorem for periodic so-lutions of autonomous delay differential equations thatstates: If an approximate frequency and periodic solu-tion is computed that satisfies a noncriticality condition,then there is an exact frequency and periodic solution ina neighborhood of the approximate frequency and periodicsolution with a specific numerical bound. To obtain thisnumerical bound requires computing several parameters.Algorithms for computing these parameters and an appli-cation will be given.

David E. GilsinnMathematical and Computational Sciences DivisionNational Institute of Standards and [email protected]

CP38

Bifurcation Analysis of Dimensionless EquationsModelling Semiconductor Lasers

The Yamada equations are ordinary differential equationsgoverning the dynamics of semiconductor lasers. It is pos-sible to use chaotic self-pulsating lasers to mask a signalin a chaotic communication system. We optimise param-eters via a symbolic and numerical bifurcation analysis ofdimensionless equations derived from the Yamada equa-tions and other models, including terms representing themodulation required to induce chaos, spontaneous emis-sion, diffusion and radiative/non-radiative recombinationeffects. The level of chaos is quantified via Lyapunov ex-ponents.

Gareth RobertsUniversity of Wales Bangor, United Kingdom%[email protected]

Ricki P. WalkerUniversity of Wales Bangor, United [email protected]

CP38

Pulsating External Cavity Semiconductor Lasers

We investigate numerically and analytically a simple ODEmodel that captures key characteristics of external cavitysemiconductor lasers. This model has the same set of fixedpoints than the ones derived from the celebrated Lang andKobayashi single mode single delay rate equations. Pul-sating phenomena originally found numerically at the levelof the Lang and Kobayashi model are investigated analyti-cally in the context of the simple ODE model coupled withthe observation that the free running semiconductor laseris a nearly conservative oscillator.

Vassilios Kovanis, Tamas WiandtRochester Institute of [email protected], [email protected]

CP38

Bifurcations of Competing Modes in Laser Arrays

We study dynamics of laser arrays in terms of composite-cavity modes for the entire coupled-laser structure. Wefind that essentially isolated micro-cavity semiconductorlasers may drive each other chaotic. As the optical iso-lation between these lasers reaches practically attainablelimits, instead of approaching independent operation, thelasers exhibit mutually induced chaotic oscillations. Themultimode equations are analyzed with bifurcation contin-uation techniques to reveal the formation of a gap in thelockband that is gradually occupied by instabilities of limitcycles.

Sebastian WieczorekSemiconductor Material and Device SciencesSandia National [email protected]

Weng ChowSandia National [email protected]

CP39

Linear Stability of Viscoelastic Shear Flows

We present a linear stability analysis of a viscoelasticOldroyd-B fluid in a plane Couette geometry. For vorticesaligned with the mean flow we find that the amplitudes ofvortices and streaks decay. At the same time, non-normalamplification of streaks leads to a transient growth, suchthat the maximum value of the kinetic energy associatedwith the perturbation exceeds its initial value by severalorders of magnitude. For Newtonian fluids the maximumamplification occurs when the width of the vortices is of theorder of the channel height. We find that for viscoelasticfluids the optimal vortex width is significantly wider andthat the efficiency of the non-normal self sustaining processis reduced.

Juergen BuehrlePhilipps-University [email protected]

Bruno EckhardtPhilipps Universitat MarburgFachbereich [email protected]

CP39

Interactions Between Polymer Dynamics and Self-Sustaining Coherent Flow Structures: a Model forTurbulent Drag Reduction

Traveling-wave solutions to the Navier-Stokes equationshave been found that capture the dominant structure ofthe near-wall buffer region of turbulence. We describe theeffect of viscoelasticity on these states, aiming to betterunderstand the mechanism(s) of drag reduction by poly-mer additives. The changes to the velocity fields mirrorthose experimentally observed in turbulent flows of poly-mer solutions: drag decreases, streamwise velocity fluctua-tions increase while wall-normal fluctuations decrease, andsmaller wavelength structures are suppressed.

Wei LiDept. of Chemical EngineeringUniversity of Wisconsin-Madison

94 DS05 Abstracts

[email protected]

Michael D. GrahamDept. of Chemical and Biological EngineeringUniversity of [email protected]

CP39

Discrete Element Modeling of Shear Bands in Soils

Soils and other granular materials can be modeled either asa continuum or as discrete particles. The second approachis known as molecular dynamics or as a discrete/distinct el-ement method (DEM). We simulate soil under several testscenarios with a discrete element method, with the goalof determining which constitutive laws are realistic with-out sacrificing computational speed. Of particular interestis which constitutive laws can generate observable shearbands, even under continuous boundary conditions.

Robert E. WiemanUniversity of BristolDepartment of Engineering [email protected]

CP40

Homoclinic Orbits in the Chua’s Circuit

We describe the complex scenario formed by the homoclinicorbits, to the saddle-focus points, in the parameter spaceof the Chua’s circuit. In particular, we present a scalinglaw that gives the ratio between bifurcation parameters ofdifferent nearby orbits. The results to be presented arevalid for a large class of dynamical systems.


Rene Medrano-TInstitute of PhysicsUnuversity of Sao [email protected]

Murilo BaptistaInstitut fuer Physik Am Neuen PalaisUniversitaet [email protected]

CP40

Chaotic Saddles and The Kuramoto-SivashinskyEquation

This work presents a methodology to study the role playedby nonattracting chaotic sets called chaotic saddles inchaotic transitions of high-dimensional dynamical systems.Our methodology is applied to the Kuramoto-Sivashinskyequation, a reaction-diffusion partial differential equation.We describes a novel technique tah uses the stable mani-fold of a chaotic saddle to characterize the homoclinic tan-gency responsible for an interior crisis, a chaotic transitionthat resultis in the elargement of a chaotic attractor. Thenumerical techniques explained here are important to im-prove the understanding of the connections between low-dimensional chaotic systems and spatiotemporal systems

which exhibit temporal chaos and spatial coherence.

Elbert E. MacauLAC - Laboratory for Computing and AppliedMathematicsINPE - Brazilian Institute for Space [email protected]

Erico RempelDGEINPE - Brazilian Institute for Space [email protected]

CP40

Homoclinic Snaking in Reversible Systems

We study the unfolding of heteroclinic cycles between equi-libria in reversible systems of ODEs. It is shown that ifone of the equilibria is a saddle-focus, snaking curves ofhomoclinic orbits can be found. Along a snaking curveinfinitely many fold bifurcations of homoclinic orbits oc-cur; the corresponding solutions spread out and developmore and more oscillations about their centre. The analy-sis is illustrated by computations for a system of Boussinesqequations.

Thomas WagenknechtBristol Laboratory for Advanced Dynamics [email protected]

Juergen KnoblochTU IlmenauInstitute of [email protected]

CP41

Dynamics Under Random Boundary Conditions

Scientific and engineering systems are usually subject touncertainty or random influence. Often, the noise acts onnonlinear systems at physical boundary. Randomness canhave delicate impact on the overall evolution of such sys-tems. Taking stochastic effects into account is of centralimportance for the development of mathematical models ofcomplex systems under uncertainty. The speaker presentsrecent results on techniques for understanding dynamicalimpact of random boundary conditions.

Jinqiao DuanIllinois Institute of [email protected]

CP41

Boundary Conditions For A Substrate Binding ToAn Enzyme

We seek to construct boundary conditions for the diffusionequation in one dimension corresponding to absorption atan endpoint with the constraint that only one diffuser canoccupy the absorbed site at a time. Consequently, we con-sider a set of N diffusers which are independent except forthe constraint at the absorbing boundary. This problempotentially has applications to many systems on a molec-ular scale, including the problem of a substrate binding toan enzyme.

Mark F. SchumakerDepartment of Mathematics

DS05 Abstracts 95

Washington State [email protected]

CP42

Molecular Vibrations and the Rotating EckartFrame

By changing its shape while conserving angular momen-tum, a polyatomic molecule can return to its initial shapewith a different orientation (as a “falling cat” or a diver cando). Classical molecular dynamics is described by a Hamil-tonian dynamical system. Using geometric mechanics, thenet angle of overall rotation is explicitly described in termsof “internal” coordinates, including Jacobi coordinates, in-teratomic distances, and generalized Eckart coordinates.The net angle of overall rotation is the sum of a dynamicphase plus a geometric phase. The latter is also describedin terms of a gauge potential or, when available, molecularrotational constants.

Florence J. LinUniversity of Southern [email protected]

CP42

Symmetry and Stability in Hamiltonian Systems

Given a periodic solution to a Hamiltonian system withspecial forward or time-reversal symmetry, useful reduc-tions exist which greatly simplify the associated linear sys-tem about the periodic orbit. This has nice applicationswhen computing the Floquet multipliers to determine lin-ear stability. We demonstrate these techniques on the fa-mous figure-eight orbit of the three-body problem, reducingstability calculations down to a 2× 2 matrix whose entriesdepend on approximating the linearized system over thefirst twelfth of the orbit. Other applications to the n-bodyproblem will be presented.

Gareth E. RobertsDept. of Mathematics and C.S.College of the Holy [email protected]

CP42

Motion of Unstable N-Ring Vortex Points on aSphere with a Background Flow

We consider the polygonal ring configuration of N identi-cal vortex points, say N-ring, on a sphere in the presenseof a background flow. Starting with the linear stabilityanalysis, we investigate unstable and recurrent motions ofthe perturbed N-ring from the viewpoint of the theory ofdynamical system using a projection method.

Takashi SakajoHokkaido UniversityDepartment of [email protected]

CP43

A Complex Network of Scientific Collaborations ofthe V European Framework Program

A collaboration network among companies and EuropeanUniversities of the V Framework Program is thoroughly an-alyzed. When this network is represented by a graph, weobtain valuable information about how science and tech-

nology are related. Moreover, we demonstrate that thisgraph is ’scale-free’ with a high clustering coefficient, show-ing some kind of preferential attachment. In addition,the presence of Universities is important in the transferof knowledge, despite they represent only a small fractionof all participants.

Miguel A. SanjuanNonlinear Dynamics and Chaos GroupUniversidad Rey Juan [email protected]

Juan AlmendralNonlinear Dynamics and Chaos GroupUniversidad Rey Juan Carlos, Madrid, [email protected]

Joao GamaDepartment of PhysicsUniversidade de Aveiro, Aveiro, [email protected]

Luis LopezLaboratory of Distributed AlgorithmicsUniversidad Rey Juan Carlos, Madrid, [email protected]

Jose MendesDepartment of PhysicsUniversidade de Aveiro, Aveiro, [email protected]

CP43

A Local Method for Detecting Community Struc-ture in Networks.

We propose a novel method of community detection that isboth computationally less complex than many competingtechniques, O(n), while possessing more physical signifi-cance to a member of, for example, a social network. Inaddition, this method is truly local: a community can bedetected within a network without requiring knowledge ofthe entire network, by a directed crawler. Several networks,including the famous Zachary Karate Club, are analyzed.


James P. BagrowClarkson UniversityDepartment of [email protected]

CP43

Statistics of Cycles: How Loopy Is Your Network?

We study the distribution of cycles of length h in large net-works and find it to be an excellent ergodic estimator, evenin the extreme inhomogeneous case of scale-free networks.The distribution is sharply peaked around a characteristiccycle length h∗ ∼ Nα. We present a Monte-Carlo sam-pling algorithm for approximately locating h∗. We showthat for small random scale-free networks of degree expo-nent λ, α = 1/(1 − λ), and α grows as the net becomeslarger.

Erik Bollt, Joseph Kirk, Daniel ben-Avraham,

96 DS05 Abstracts

Hernan D. RozenfeldClarkson [email protected], [email protected],[email protected], [email protected]

CP44

Pattern Selection for Impulsively-Forced FaradayWaves

We consider parametrically-excited standing waves on afluid subjected to a periodic sequence of delta-function im-pulses, an idealization of the Faraday system for whichwaves are excited by sinusoidal modulation of gravity. Us-ing the Zhang-Vinals model of weakly-viscous Faradaywaves, we construct an explicit stroboscopic map that de-termines the weakly nonlinear evolution of a class of two-dimensional wave patterns. We compare our pattern selec-tion results with those obtained numerically for sinusoidaland multi-frequency forcing.

Anne J. CatllaNorthwestern [email protected]

Mary C. SilberNorthwestern UniversityDept. of Engineering Sciences and Applied [email protected]

CP44

Analysis of the Nonlinear Behavior of the ForcedResponse of Combustion Systems

We present results from analysis of data from two com-bustion experiments conducted in which harmonic modu-lation of fuel flow was employed as a means to improvecombustion stability. We investigated the observed non-linear interaction between the driven oscillations and thenatural acoustic modes of the combustion chamber. Thedynamic properties of the forced oscillations were analyzedwith time delay embedding techniques. For the low pres-sure combustion experiment, fluctuations of the attractorand a boundary crisis occurred near instability. For thehigh pressure experiment, various regimes of quasiperiod-icity, multistability, various modes of synchronization, andchaotic behavior were observed as the forcing frequencywas varied.

Nikolai F. RulkovInformation Systems [email protected]

Michael L. LarsenInformation Systems [email protected]

CP44

When Noise Influences the Collapse of Spatiotem-poral Chaos

Wave-induced spatiotemporal chaos in the Gray-Scottmodel, an excitable medium based on cubic autocataly-sis, is transient with an exponential increase of the aver-age transient time with medium size. The collapse of thisdiffusion-sustained spatiotemporal chaos is initiated by in-trinsic statistical spatial correlations that force the mediuminto the asymptotically stable steady state. The robustnessof the collapse process in the presence of spatially homoge-

neous as well as spatially inhomogeneous dynamical noise- ubiquituous in real systems - is investigated.We find thatthe collapse process is delayed in the presence of small-amplitude inhomogeneous dynamical noise, but advancedby spatially homogeneous noise and spatially inhomoge-neous large-amplitude noise.

Sumire KobayashiDartmouth CollegeDepartment of [email protected]

Renate A. WackerbauerUniversity of Alaska FairbanksDepartment of [email protected]

CP45

Metastable States and Modelling of PersistantNeural Activity

Starting from the Kolmogorov equation, we describe dy-namics of the mean and the variance of neural activity. Bi-furcation analysis shows that a metastable state (the meanis bounded and the variance grows to infinity) exists in thesystem. Basing on this study, we simulate a neural net-work of integrate-and-fire elements with noise and applythe result of simulations to describe the experimental dataon persistent activity in the brain.

Roman M. BorisyukUniversity of PlymouthCentre for Theoretical and Computational [email protected]

CP45

N-Bump Solutions of Amari-Type Equation

We explore a partial integro-differential equation (Amari,Troy et.al.) that models pattern formation in neuronalnetworks:

∂u(x, t)

∂t= −u(x, t)+

∫ ∞

−∞ω(x−y)f(u(y, t))dy+s(x, t)+h.

The main objective has been to establish the existence andstability of N-bump stationary solutions. Our focus hasbeen to extend existing results for the equal width bumpcase, establish stability of those solutions, and characterizea class of mexican-hat coupling functions that allow N-bump solutions.

Fernanda Botelho, James Jamison, Angie MurdockUniversity of [email protected], [email protected], [email protected]

CP45

Dynamics of Fluctuation-Driven Neuronal Net-works

Temporal fluctuations in the synaptic coupling betweenneurons naturally arise in neuronal networks and canhave a significant effect on the dynamics of the network.These fluctuations are especially important when the meansynaptic drive of individual neurons are not sufficient toevoke spiking response. To highlight the importance ofthese fluctuations, we contrast the dynamics in these net-works with mean-driven networks where the mean synap-

DS05 Abstracts 97

tic input into each neuron is sufficiently strong to drivespiking. We find that mean-driven networks tend to behysteretic and harder to control and that the activation ofcertain neuroreceptors (e.g., NMDA or AMPA) can deter-mine the nature of the network dynamics.

Louis TaoDepartment of Mathematical SciencesNew Jersey Institute of [email protected]

CP46

Application of General Models in Model Aggrega-tion

The modeling of complex systems often leads to modelswith high numbers of state variables and parameters. Suchmodels are difficult to study. Moreover, the high number ofparameters means that many experimental measurementsare necessary for tuning and validation. It is therefore of-ten desired to reduce the complexity of a given model. Wepresent a new approach for model aggregation which isbased on the application of general models. In contrast toothers this approach conserves the local bifurcation struc-ture of the system.

Ulrike FeudelICBM Theoretical Physics/Complex SystemsCarl von Ossietzky Universitaet [email protected]

Thilo GrossFachbreich PhysikUniversitat [email protected]

Dirk StiefsICBM, Theoretical Physics/Complex SystemsCarl von Ossietzky University Oldenburg, [email protected]

CP46

Reduced Atmospheric Models: Proper Basis Func-tions, Dimensionality, Stochastic Modeling of Fast-Evolving Modes

The construction of reduced atmospheric models, that is,models that explicitly deal only with a limited number ofessential degrees of freedom while keeping as much realismas possible has attracted some attention in recent years. Inthe present paper, nonlinear reduced models of large-scaleatmospheric dynamics are derived using a quasigeostrophicthree-level spectral model with realistic variability as dy-namical framework. The study focuses on three issues: (i)finding appropriate basis functions for efficiently spanningthe dynamics and a comparison between different choices ofbasis functions; (ii) the minimal dimension of the reducedmodel necessary to faithfully simulate certain aspects of thelong-term behavior of the full spectral model; (iii) the ques-tion of whether the influence of unresolved fast-evolvingmodes onto the resolved slowly-evolving large-scale modescan be modeled by stochastic terms.


CP47

Unsteady Fluid Flow Separation by the Method ofAveraging

We use the method of averaging to improve recent separa-tion criteria for two-dimensional unsteady fluid flows withno-slip boundaries. Our results apply to general compress-ible flows that admit a well-defined asymptotic average.Such flows include periodic and quasiperiodic flows, as wellas aperiodic flows with a mean component. As an exam-ple, we predict and verify unsteady separation location andangle in variants of an oscillating separation bubble model.

George HallerMassachusetts Institute of TechnologyDepartment of Mechanical [email protected]

Mustafa Sabri KilicDepartment of MathematicsMassachusetts Institute of [email protected]

Anatoly NeishtadtSpace Research InstituRussian Academy of [email protected]

CP47

Experimental Investigations of a Dynamical Sys-tems Approach to Unsteady Separation

We present the results of a study concerning the applica-tion of a new approach to unsteady flow separation in anexperimental setting. The flow geometry used is the ‘rotor-oscillator’ flow, which allows for investigation of separationunder carefully-controlled unsteady flow conditions. Thelocation and geometry of the separating material spike arereported for qualitatively different unsteady flows, and theresults related to recent theoretical work concerning fixedand moving separation.


Thomas Peacock, Raul [email protected], [email protected]

CP47

Kinematic Theory of Unsteady Separation in ThreeDimensional Flows

Exact quantitative characterization of separation in threedimensional unsteady flows over no-slip boundaries hasbeen an outstanding problem. We attribute separationto the existence of non-hyperbolic invariant manifolds at-tached to the boundary. With this viewpoint we derive nec-essary and sufficient conditions for existence of these mani-folds and their shape using: Normally hyperbolic invariantmanifolds, Distinguished asymptotic behavior of materialsurfaces/lines, Averaging and Finite Time Invariant Man-ifolds. We verify these conditions on different examples.

George HallerMassachusetts Institute of TechnologyDepartment of Mechanical Engineering

98 DS05 Abstracts

[email protected]

Amit Surana, Olivier GrunbergDepartment of Mechanical EngineeringMassachusetts Institute of [email protected], [email protected]

CP48

Symmetry of Attractors and Frobenius-Perron Op-erator

The application of group representation theory for ana-lyzing the Frobenius-Perron (F-P) operator of equivariantmaps (both continuous maps and homeomorphisms) is dis-cussed. We consider applications to admissible symmetrytypes of attractors, symmetry increasing bifurcations of at-tractors (via collisions of attractors) as well as – from thecomputational point of view – to the efficient discretiza-tion of the F-P operator in symmetry adapted co-ordinates,computations of invariant measures, and to the detectionof bifurcations of attractors.

Mirko HesselInstitute for MathematicsUniversity of [email protected]

Prashant G. MehtaUnited Technologies Research [email protected]

Michael DellnitzInstitute for MathematicsUniversity of [email protected]

CP48

Accurate Computation of Invariant Densities

Invariant densities, when they exist, describe the statisticalbehaviour of a dynamical system. Ulam’s method (involv-ing a discretization of a transfer operator) can be used toapproximate such densities numerically, but convergencecan be very slow when the density has complicated struc-ture (for example, in the logistic family of one-dimensionalmaps). In this talk I’ll present a strategy for automatic par-tition selection in Ulam’s method which accelerates conver-gence and yields surprisingly sharp images. The behaviourof the algorithm also suggests a phenomenological test forchaos.

Rua MurrayDepartment of MathematicsUniversity of [email protected]

CP48

A Set Oriented Approach for the Approximation ofInvariant Manifolds Using Finite-Time LyapunovExponents

Invariant manifolds determine the geometric skeleton of adynamical system and play a crucial role in transport andmixing processes. In nonautonomous systems, often sta-tistical methods such as finite-time Lyapunov exponents(FTLE) are used to pinpoint the geometrical structuresof interest, exploiting that the stable manifolds of hyper-bolic objects are typically local maximizers of the FTLE.

Based on these results we present a set oriented approachfor the approximation of invariant manifolds and illustratethe methods by several examples.

Michael Dellnitz, Kathrin PadbergUniversity of Paderborn, [email protected], [email protected]

CP49

Pump Coupled Lasers with Delay

We considered two lasers coupled by modifying the pumpsignal of one laser by the light-intensity fluctuations of theother. The optical-to-electronic feedback can lead to asignificant time delay in the coupling signal. We inves-tigate how the coupling strength and delay time affect thesynchronization of the two lasers and observe a surprisingweak-coupling resonance effect. We compare our results toan experimental system of two cross-coupled semiconduc-tor lasers.

Thomas W. CarrSouthern Methodist UniversityDepartment of [email protected]

Ira B. SchwartzNaval Research LaboratoryNonlinear Dynamical Sysytems [email protected]

CP49

Delay Differential Equations Modeling Lasers

Lasers subject to optical or opto-electronic feedback ex-hibit pulsating outputs that are described in terms of twoor three delay differential equations. Bifurcations to peri-odic and quasi-periodic oscillations, as well as the isolatedbranching of steady and periodic solutions can be deter-mined analytically. A series of basic problems for the sim-ple and multiple Hopf bifurcations are reviewed and theirrelevance for other areas of science and engineering is dis-cussed.

Thomas ErneuxUniversite Libre de BruxellesOptique Nonlineaire [email protected]

CP49

Dynamics of Fiber Laser Arrays

Recent experiments show that fiber lasers can be synchro-nized simply by coupling them through an optical waveg-uide near the output end. These results can lead to dra-matic improvements in technology, yet how these laserssynchronize is not understood. We propose a theoreticalmodel based on an iterated map with a particular sym-metry. Our model captures key qualitative features seenin the experiments and provides an insight into underlyingdynamics.

Kurt WiesenfeldGeorgia TechDepartment of Physics

Slaven PelesGeorgia Institute of Technology

DS05 Abstracts 99

[email protected]

Jeffrey RogersHRL Laboratories [email protected]

CP50

Localized Elastic Instabilities and WKB-typemethods

In this this work we show how localised responses arise inthe context of non-homogeneous linear bifurcation equa-tions for several structural models; The WKB method withturning points is employed to locate localized instabilitypatterns, and then numerical simulations are performedto establish the correctness of our asymptotic analysis. Incontrast to similar works on structural localization, the ap-proach we have taken seems to be more natural in dealingwith localization, as it is based on the concept of turn-ing points and which in solid mechanics have a naturalinterpretation as points which are higly stressed. This cor-responds to the physical intuition that localization is di-rectly related to those parts of the structure experiencingthe most severe deformation.

Ciprian D. ComanUniversity of Glasgow, Department of [email protected]

CP50

On Control As a Facet of Mechanics

The topic of control may be considered naturally in thecontext of the physical theories of the potential and dis-sipation. In this setting, the control design process maybe thought of as the compilation of potential reconstruc-tion and dissipation remodeling. In fact, so far as sim-ple mechanical systems are concerned, the commonly usedtechniques are precisely of this character. In this talk thisphilosophy and its extensions are addressed.

Gregory P. Hicks

NRC/[email protected]

CP51

Bifurcations and Transient Response of a DynamicBalancer for Rotating Machines

We present a detailed analysis of the bifurcations and tran-sient response of a dynamic balancer for rotating machines.Continuation techniques are employed to compute curvesof equilibria and their bifurcations as parameters are var-ied. A stable balanced state is identified for a broad rangeof physical parameters. However, complex unbalancedstates including limit-cycles and chaotic motion are alsofound. Transient response and sensitivity of the balancedstate under perturbation are analysed using pseudospectratechniques.


Kirk GreenUniversity of BristolBristol Laboratory for Advanced Dynamics Engineering

[email protected]

Mike Friswell, Nick LievenAerospace EngineeringUniversity of [email protected], [email protected]

CP51

Parasitic Solutions of a Discrete Rod Model

Analysis of a discrete model of the unshearable and inex-tensible Kirchoff-rod, consisting of straight rods and elas-tic joints will be presented. The discretization makes therod shape non-unique, and causes parasitic solutions dur-ing the computation of the equilibrium branches of twistedrings. Those parasitic solutions do not satisfy the closingconditions. We will present an efficient way of labeling thejoints, what makes the rod shape unique, and allow us thecomputation of equilibrium branches.

Robert NemethDepartment of Theoretical and Applied MechanicsCornell [email protected]

CP51

Stability of a Whirling Conducting Rod in thePresence of a Magnetic Field: a Model for a SpaceTether

We study whirling instabilities of a conducting rod whirlingin a magnetic field by performing a (largely numerical) con-tinuation and bifurcation analysis. Attention is also paidto exact finite-length helical solutions for which exact re-sults can be obtained. The results are relevant for theShort Electrodynamic Tether (SET) prototype of the Eu-ropean Space Agency, which is designed to use the earth’smagnetic field, rather than chemical fuel, for thrust anddrag.

Gert van der HeijdenCentre for Nonlinear DynamicsUniversity College [email protected]

Juan ValverdeDepartment of Mechanical and Materials EngineeringUniversity of [email protected]

CP52

Numerical Study of Stochastic Reaction-DiffusionEquations Subject to Internal Noise

We address the problem of implementing numerical al-gorithms to study models of reaction-diffusion problemssubject to internal noise. The schemes proposed conservethe nonnegativity of the solutions and allow us to inves-tigate the effect of small perturbations in the propagationof fronts in reaction-diffusion problems. We also study thedevelopment of instabilities due to the discretenes of par-ticles and the statistical properties of front interfaces inhigher dimensions.

Esteban MoroDepartamento de MatematicasUniversidad Carlos III de [email protected]

100 DS05 Abstracts

CP52

Determination of Barriers in Bistable StochasticDynamical Systems A Graph Theoretic Approach

Understanding global stochastic dynamics depends on de-termination of pseudo-barriers with transport only due tostochastic basin hopping. This is particularly true formulti-stable systems, the key being to find closest ap-proaches between stable and unstable manifolds of thebasins. We take a graph theoretic approach, by approxi-mately projecting the Frobenius-Perron operator onto basisfunctions, and then determining communities correspond-ing to basins, We develop techniques and language fromrecent advances in random graph theory.

Naratip SantitissadeekornClarkson [email protected]

CP53

A Model for Human Ventricular Action Potentials:Insights into Brugada Syndrome Mechanisms

Using published experimental data, we have developed cellmodels of the Brugada syndrome, a genetic disease thatcan cause sudden cardiac death without underlying struc-tural defects. Because Brugada syndrome affects cells indifferent regions of the ventricles differently, we also havedeveloped models of different cell types present in humanventricles as well as a spatial representation of their distri-bution through the ventricular wall. In this talk we presentpotential mechanisms by which Brugada syndrome maylead to lethal arrhythmias.

Victor Perez-GarciaUniversidad de Castilla-La [email protected]

Steven J. EvansBeth Israel Medical CenterNew York, [email protected]

Flavio FentonBeth Israel Medical Center & Hofstra UniversityNew York, [email protected]

Elizabeth M. CherryHofstra UniversityHempstead, [email protected]

Alfonso BuenoDepartment of MathematicsUniversidad de Castilla-La [email protected]

CP53

Fibrillation without Alternans in Porcine Ventri-cles: Theory and Numerical Simulations

Despite simple restitution theory that predicts electricalalternans and fibrillation for restitution curve slope¿1,porcine ventricles exhibit steep restitution without alter-nans. With numerical simulations using a mathematicalmodel for cell electrophysiology and a 3D anatomical modelof porcine ventricles, we explain how specific properties of

pig action potential shape can produce large electrotoniceffects that suppress alternans despite slope¿1 .As in ex-periments we obtain fibrillation by an extra stimulus butnot by fast pacing.

Richard GrayDepartment of Biomedical EngineeringUniversity of Alabama at [email protected]

Steven EvansBeth Israel Medical [email protected]

Harold M. HastingsHofstra UniversityDept. of Physics - CHPHB [email protected]

Elizabeth M. CherryHofstra UniversityPhysics [email protected]

Flavio FentonBeth Israel Medical Center & Hofstra UniversityNew York, [email protected]

CP53

Calcium Alternans and Intracellular Calcium Cy-cling in Cardiac Cells

Calcium (Ca2+) alternans in cardiac cells are beat-to-beatalternations in the amplitudes of the systolic Ca2+ tran-sient. We will talk about the mechanisms by which in-tracellular Ca2+ cycling induces Ca2+ alternans. First,we set up a discrete time Ca2+ movement model withtwo compartments: cytoplasm and sarcoplasmic reticulum(SR). The model shows that reduced SR Ca2+ release canincrease the SR content. When the SR content reachesa threshold, CICR becomes unstable and Ca2+ alternansare exposed. A primary mechanism for this instability isthe steep, nonlinear SR Ca2+ release function. Second,we use a CICR model by Keizer and Smith (1998) to ex-plore the relation between Ca2+ waves and Ca2+ alter-nans. We applied a local periodic Ca2+ stimulation withvarying pacing interval (PI). When the total Ca2+ is high(Ca2+ overload), the local Ca2+ stimulation with the largePI induces Ca2+ waves (1:1 rhythm). As PI is reduced,propagated Ca2+ waves alternate with waves that fail topropagate, giving a 2:2 rhythm. This results from slowedrecovery from refractoriness of the SR Ca2+ channel at thereduced PI.

James P. KeenerUniversity of [email protected]

Young-Seon LeeDepartment of MathematicsUniversity of [email protected]

CP54

The Impact of Behavior Changes on the Spread of

DS05 Abstracts 101

a Smallpox Epidemic

We propose and analyze a mathematical model to considerthe impact that behavior changes can have spread of small-pox. We assume that some individuals will lower their dailycontact activity rates once an epidemic has started anddemonstrate that the spread of the disease is highly sen-sitive to how rapidly the population reduces their contactactivity rates. We analyze the effectiveness of mass vacci-nation and quarantining the infected population throughcontact tracing.

Sara Del ValleLos Alamos National LaboratoryLos Alamos National [email protected]

Carlos Castillo-ChavezDepartment of Mathematics and StatisticsArizona State [email protected]

Mac HymanLos Alamos National [email protected]

Herb HethcoteUniversity of [email protected]

CP54

Multiscale Stochastic Approaches for Spatiotempo-ral Disease Spread

Richard JordanHealth and Enironmental Security GroupDynaimcs Technology, [email protected]

CP54

Cell-Centered Computational Modeling of DeNovo and Sprouting Blood Vessel Growth

An essential question for developmental biology is howcells shapes and behaviors drive their assembly into tis-sues. Experimental studies have extensively analyzed en-dothelial cell behavior during blood vessel development.We integrate such experimental data using a synthetic ap-proach which constructs computer models mimicking en-dothelial cell behavior. In the first stages of blood ves-sel growth endothelial cells form a network-like structure,called the primary capillary plexus. The plexus loses un-derused branches, expands and remodels by associatingwith additional cell types, transforming itself into a vas-cular tree. Capillary plexi can form either through vascu-logenesis, the assembly of disconnected endothelial cells, orthrough angiogenesis, the sprouting or subdivision of ex-isting blood vessels. Many different computational modelshave attempted to explain and describe both vasculogene-sis or angiogenesis. Since vasculogenesis and angiogenesisare closely related processes regulated by the same geneticmachinery, a plausible mechanism must explain both. Ourstrategy for elucidating how endothelial cells form vascularnetworks and how new vessels sprout from existing ves-sels, is to focus on individual cell behavior. We aim toreconstruct the minimal set of cell behaviors that sufficesfor vascular patterning. Using the Cellular Potts model-a cellular-automaton-based Monte-Carlo technique meso-

scopically simulating cell motility via membrane extensionsand retractions-we have built a computational model of invitro endothelial cell cultures. The model quantitativelyreproduces in vitro vasculogenesis and subsequent in vitroremodeling, and also reproduces aspects of sprouting angio-genesis. Our models predict that cell polarization, throughthe elongation of the endothelial cells, and the rate of dif-fusion of a morphogen are key to correct spatiotemporalin silico replication of vascular patterning and subsequentremodeling.

Roeland MerksIndiana UniversityBiocomplexity [email protected]

CP55

Stochastic Phase Resetting of Coupled Ensemblesof Phase Oscillators Stimulated at Different Times.

We study the transient resynchronization dynamics of twoweakly interacting ensembles of noisy phase oscillators af-ter being phase reset at different times. Different couplingstrengths and distribution of eigenfrequencies are studied.We analyze the different types of mechanisms in whichthese ensembles reorganize to resynchronize. We demon-strate the impact of eigenfrequency distribution and noiseamplitude on the resynchronization behavior and its con-sequences for the design of therapeutic brain stimulationtechniques.

Jorge N. BreaCenter of Neurodynamics, University of Missouri [email protected]

Peter A. TassInstitute of Medicine (MEG)Research Centre [email protected]

Frank E. MossCenter for NeurodynamicsUniv. of Missouri - St [email protected]

Kevin DolanInstitute of Medicine, Research Center Juelich, [email protected]

CP55

Adaptive Synchronization of Coupled NonlinearOscillator Arrays Subject to External Forcing

We present results on the response of a system of coupledvan der Pol oscillators to inhomogeneous periodic externalforcing. We describe a scheme to make the Arnold Tongueadaptive with regard to variations in the external forcingamplitude and frequency by changing the array parame-ters. The method is also applied to the same system withadded external noise. This has application to signal pro-cessing using nonlinear oscillator arrays.

Lesley Ann LowInformation Systems Laboratories, [email protected]

102 DS05 Abstracts

CP55

Phase Synchronization and Coherence Analysis:Sensitivity and Specificity

When applying multivariate time series analysis techniquesto empirical data, conclusions about the underlying dy-namics generating the data are of particular interest.Therefore, not only sensitive analysis techniques for investi-gating direct problems but also specific analysis techniquesare desired to avoid erroneous conclusions in inverse prob-lems. We illustrate the problem of missing specificity ofphase synchronization and coherence analysis. We proposea methodology to increase specificity and demonstrate itsperformance using two dynamic model systems.

Jens TimmerUniversity of FreiburgDepartment of [email protected]

Matthias WinterhalderCenter for Data Analysis and ModellingUniversity of Freiburg, [email protected]

Bjorn SchelterFreiburg Center for Data Analysis and ModelingUniversity of [email protected]

Juergen KurthsPhysics DepartmentUniversity of [email protected]

MS1

Fluctuation-Induced Pattern Formation in a Sur-face Reaction

Nucleation, pulse formation and subsequent propagationfailure have been observed in the CO oxidation on Pt(110)at intermediate pressures (≈ 10−2 mbar). This phe-nomenon can be reproduced with a stochastic model in-cluding temperature effects, in which nucleation occurs viafluctuations, whereas the subsequent behaviour follows es-sentially the deterministic path. Conditions for the obser-vation of stochastic effects in the pattern formation duringCO oxidation are discussed.

Jens StarkeUniversity of HeidelbergInstitute of Applied [email protected]

Markus EiswirthFritz-Haber-Institute, [email protected]

Christian ReichertUniversity of [email protected]

MS1

Reactive Phase Separation on Catalytic Surfaces

Energetic interactions between the reacting species in areaction-diffusion system may lead to a reactive phase sep-aration. In a Turing-like instability periodic patterns will

develop in such a case. Catalytic reactions with alkali met-als as promoters represent systems of this type as has beenshown with Rh(110)/K where in the O2+H2 reaction K+Oislands of macroscopic dimensions form. In the excitablesystem NO+H2 on Rh(110) the presence of potassium leadsto pulses transporting the alkali metal. The conditions forreactive phase separation are not at all restrictive and it isexpected that such a mechanism applies to a broad classof systems. We also discuss new aspects in the mathe-matical modeling of such systems and present numericalsimulations.

Ronald ImbihlInstitut fur Physikalische Chemie und ElektrochemieUniversity of Hannover, [email protected]

MS1

Traveling Waves in Rapidly Varying HeterogenousMedia

Traveling waves can become pinned or modulated in re-action diffusion equations in unbounded cylindrical het-erogenous media. The presence of small scale structuresin the medium can cause failure of propagation (’pinning’)so that rather than having travelling waves one has sta-tionary, spatially localized solutions. We use a descrip-tion of the elliptic equation on the unbounded domain asan (ill-posed) evolution equation together with exponentialaveraging techniques to show that pinning occurs only invery small ranges of parameters. With similar techniques,it is also possible to describe the periodic modulation oftravelling waves i.e., travelling wave solutions can be de-scribed by a spatially homogenous equation and exponen-tially small remainders. This is partly joint work with H.Uecker, G. Schneider, and C.E. Wayne.

Karsten MatthiesInstitut fur Mathematik IFree University Berlin, [email protected]

MS1

Corners in Front Propagation

We investigate corners of interfaces in anisotropic systems.Starting from a stable planar front in a general reaction-diffusion system, we show existence of almost planar inte-rior and exterior corners. When the interface propagationis unstable in some directions, we show that small stepsin the interface may persist. Our assumptions are basedon physical properties of interfaces such as linear and non-linear dispersion, rather than properties of the modelingequations such as variational or comparison principles. Wealso comment on corners in pattern forming fronts and in-terfaces between planar wave trains.

Arnd ScheelUniversity of MinnesotaSchool of [email protected], [email protected]

MS2

Multiscale Modeling and Numerical Methods forSorting of Particles in Heterogeneous Devices

The problem of separation of large biomolecules such asDNA and proteins is of high interest for biological re-search and biomedical application. In this presentation

DS05 Abstracts 103

we discuss a multiscale modeling approach for the sepa-ration of particles in heterogeneous devices such as microarrays. These transport phenomena are modeled with mul-tiscale advection-diffusion equations. This new modelingapproach allows to investigate theoretically and numeri-cally the macroscopic behaviour of the micro array andits effects on the injected particles, The numerical simula-tion of this problem, present a number of challenges due tothe multiple scales involved. We will also discuss numericalmethods for the efficient solution of such type of multiscaleequations.

Assyr AbdulleUniversity of BaselDepartment of [email protected]

MS2

Multiscale Modeling of an Underground NuclearWaste Site

An underground nuclear waste repository consist of nuclearwaste packages stored in excavated vaults, all connected bydrifts and galleries or tunnels which are backfilled after thepackages storage. Usually, for safety reasons, the entirerepository site is embedded in a low permeability layer.On the one hand, the model of a repository site should in-clude the multi scale geometry, the large variations of thegeology and the coupling of the different phenomena. . Buton the other hand far field simulations for performance as-sessments cannot use such detailed models. In the case of ahigh number of leaking packages with a possible damagedzone, we present the results of mathematical homogeniza-tion and asymptotical methods, leading to scaled up butaccurate and physical macroscopic model. Both proofs ofaccuracy and numerical simulations demonstrate the qual-ity of the scaling up.

Alain BourgeatEquipe MCSUniversite [email protected]

MS2

Numerical Homogenization of Nonlinear ParabolicDifferential Equations and its Applications

The numerical homogenization methods presented in thistalk are designed to compute homogenized solutions. Iwill describe numerical homogenization methods that weproposed recently and their relation to some other multi-scale methods. Convergence of these methods for nonlinearparabolic equations will be discussed. Numerical examplesand applications will be considered.

Yalchin EfendievDept of MathematicsTexas A&M [email protected]

MS2

Variational Multiscale Methods andFront-Tracking Techniques for Multiphase Flow inPorous Media

Multiscale phenomena are ubiquitous to flow and trans-port in porous media. The problem can be expressed interms of a pressure equation (almost elliptic) and a sys-tem of transport equations (almost hyperbolic). In this

paper, we develop a novel formulation based on a locally-conservative variational multiscale method for the pressureequation, and a front-tracking technique for the solution ofthe system of transport equations along streamlines. Theproposed method captures the fine-scale heterogeneity ona coarse grid, and provides an accurate and efficient simu-lation technique for miscible and immiscible flow in porousmedia.

Ruben JuanesDepartment of Petroleum EngineeringStanford [email protected]

MS3

Evolution of Pattern Complexity in the Cahn-Hilliard Theory of Phase Separation

Phase separation processes in compound materials can pro-duce intriguing and complicated patterns. Yet, character-izing the geometry of these patterns quantitatively can bequite challenging. In this paper we use computational al-gebraic topology to obtain such a characterization. Ourmethod is illustrated for the complex microstructures ob-served during spinodal decomposition and early coarseningin both the deterministic Cahn-Hilliard theory, as well asin the stochastic Cahn-Hilliard-Cook model. While bothmodels produce microstructures that are qualitatively sim-ilar to the ones observed experimentally, our topologicalcharacterization points to significant differences.

Marcio GameiroSchool of MathematicsGeorgia Institute of [email protected]

Thomas WannerGeorge Mason UniversityDepartment of Mathematical [email protected]

Konstantin MischaikowDepartment of MathematicsGeorgia [email protected]

MS3

Topological Feature Extraction in Cubical Grids

Cubical sets are a convenient geometric structure for rep-resenting, among others, information contained in digitalimages. Moreover, the graph of a map can also be repre-sented by a cubical set in the product space. Homology isan algebraic tool for extracting information about specificfeatures of an image or a multidimensional structure. Inthis talk we present three stages of that feature extraction:Obtaining a cubical set, generating its chain complex andcomputing its homology.

Tomasz KaczynskiUniversite de SherbrookeDepartement de [email protected]

MS3

Topological Characterization of Spatial-Temporal

104 DS05 Abstracts

Chaos

It is well established both numerically and experimen-tally that nonlinear systems involving diffusion, chemo-taxis, and/or convection mechanisms can generate com-plicated time-dependent patterns. Since this phenomenonis global in nature, obtaining a quantitative mathematicalcharacterization that to some extent records or preservesthe geometric structures of the complex patterns is dif-ficult. We illustrate a technique aimed at this problem.More precisely, using algebraic topology, in particular ho-mology, we can measure Lyapunov exponents that implythe existence of spatial-temporal chaos and suggest a ten-tative step towards the classification and/or identificationof patterns within a particular system.

William D. KaliesFlorida Atlantic UniversityDepartment of Mathematical [email protected]

MS3

Computing Homology of Maps

An algorithm computing homology of maps is needed inrigorous qualitative numerical analysis of dynamical sys-tems based on topological tools such as Conley Index,Fixed Point Index or Degree Theory. The typical rigorousinformation one can extract from an implicitly given mapare upper estimates of images of sets in the map, usuallygiven in the form of a multivalued representation. In thelecture we review two algorithms for computing homologyof continuous maps based on multivalued representationand cubical homology. The first algorithm utilizes a directconstruction of a chain selector of the multivalued represen-tation. The other algorithm is based on projections fromthe graph of the multivalued map onto the domain and theset of values. We also briefly mention a third approach,based on some ideas of Cech homology.

Marian MrozekJagiellonian UniversityInstitute of Computer [email protected]

MS4

Schroedinger Maps Near Harmonic Maps

The Schroedinger map equation is a basic model in ferro-magnetism, as well as a geometric (and hence nonlinear)version of the linear Schroedinger equation. It is an openquestion whether finite energy solutions are global, or blowup in finite time. Here we consider equivariant Shroedingermaps from 2+1-dimensional space-time into the 2-sphere,with energy close to the energy of an equivariant harmonicmap. We show that solutions stay close to harmonic mapsbefore blow up (if any), and that they blow up if and only ifthe length scale of the nearest harmonic map goes to zero.This is joint work with Kyungkeun Kang and Tai-PengTsai at UBC.

Stephen Gustafson, Kyungkeun Kang, Tai-Peng TsaiDepartment of MathematicsUniversity of British [email protected], [email protected],[email protected]

MS4

Numerical Study of Singular Solutions to the

LL(G) Equations

The Landau-Lifshitz and Landau-Lifshitz Gilbert equa-tions are the basic evolution equations in micromagnetics,a continuum model for magnetic behavior in ferromagneticmaterials. In the setting where the magnetic behavior isdetermined by the Dirichlet energy, these equations are ahybrid Schrodinger map flow and harmonic map heat flowinto the unit sphere S2. The question of singularity for-mation for finite energy data for these equations is open,but the search is motivated by the fact that under a suit-able transformation, these equations are reminiscent of thecubic nonlinear Schrodinger equation for which singularsolutions abound. Analytical attempts have met impassesbut have yielded insight which underlies current numericalinvestigations (in collaboration with S, Bartels).

Joy KoBrown [email protected]

MS4

Schrodinger Maps and theLandau-Lifshitz-Maxwell Equation

A special case of the geometric PDEs known as Schrodingermaps, the Landau-Lifshitz equation is a nonlinear PDEthat describes the magnetic moment, or spin, of a ferro-magnetic material. Since the spin contributes a magneticfield, we couple the above to Maxwell’s equations govern-ing electromagnetic fields. Using geometric ideas from ourprevious work on Schrodinger maps, along with energy es-timates for the Landau-Lifshitz-Maxwell system, we provelocal existence. However, long-term behavior is an openquestion.

Helena McGahaganUniversity of California at Santa [email protected]

Jalal ShatahNew York [email protected]

MS4

Anti-ferromagnetic Chains and Schrodinger Mapswith Strong Potentials

In the Heisenberg model, the continuum limit of the evo-lution of the spin vectors on a ferromagnetic lattice isdescribed by Landau-Lifshitz equation. While the anti-ferromagnetic chains are also described by the Heisenbergmodel, we show rigorously that the continuum limit is theσ model, i.e. the wave map targeted on the unit sphere.This result is proved in a general setting of Schroding mapswith strong potentials.

Chongchun ZengUniversity of VirginiaDepartment of [email protected]

Jalal ShatahNew York [email protected]

DS05 Abstracts 105

MS5

Self-Organization in Microbial Colonies

In nature, microorganisms must often cope with hostileenvironmental conditions. To do so they have developedsophisticated cooperative behavior and communication ca-pabilities. Utilizing these capabilities, the colonies developcomplex spatio-temporal patterns. We present a wealthof beautiful patterns formed during colony development ofvarious microorganisms. Invoking ideas from pattern for-mation innon-living systems and using “generic” modelingwe are able to account for the salient features of the obser-vations.

Eshel Ben-JacobTel-Aviv [email protected]

Herbert LevineUniv. of Cal. at San DiegoDepartment of [email protected]

MS5

Motion of Biological Organisms as Interacting Self-propelled Particles

Organisms, from cells to humans, move and interact. Tounderstand the emerging order, models consisting of self-propelled particles can be studied. Early results showedthat local alignment can order the motion of the wholegroup, even in the presence of noise. Slightly more intel-ligent particles can describe regularities in pedestrian mo-tion, such as spontaneous lane formation or panic. Motionand interaction between tissue cells is required for the for-mation of anatomical structures, as seen in the example ofthe vascular network.

Andras Czirok, Illes FarkasDepartment of Biological PhysicsEotvos [email protected], [email protected]

Tamas VicsekDepartment of Biological PhysicsEotvos [email protected]

MS5

Modelling Daphnia Swarming

We propose a self-propelled particle model for the swarm-ing of Daphnia, which takes into account propulsion of theparticles, mutual avoidance of close encounters and attrac-tion to a center. Various key parameters are identified inorder to arrive at a phase diagram for qualitatively differ-ent steady-state motions. We find that a vortex is formedonly in a finite range of propulsions, and analyze its tran-sitions to other states. Hydrodynamic interaction betweenthe particles can stabilize the vortex and change its velocityprofile.


MS5

A Nonlocal Continuum Model for Localized Bio-logical Aggregations

We construct and study a nonlinear, nonlocal continuummodel for the movement of biological populations whosemembers experience long-range social attraction and short-range dispersal. Using phase plane analysis, energy meth-ods, and numerical computations, we study the dynamics,pattern selection, and steady states. The model displayscoarsening behavior, and has localized, clump-like steadysolutions with key characteristics observed in natural bi-ological aggregations, namely sharp boundaries and con-stant internal population density.

Mark LewisUniversity of Alberta, [email protected]

Chad M. TopazUCLADepartment of [email protected]

Andrea BertozziUCLA Department of [email protected]

MS6

The Effect of Feedback on the Pacemaker Unit ofa CPG

In certain CPGs, pacemaker units have inhibitory synapsesonto other neurons which display short-term synaptic plas-ticity. In this talk, we discuss the ramifications of synap-tic feedback onto the pacemaker. Feedback changes thefrequency of the pacemaker, which, in turn, affects thestrength of its synapses. Using geometric singular pertur-bation theory, we analyze how feedback inhibition affectsthe phase relationship between neurons in the pyloric net-work CPG.

Amitabha K. BoseNew Jersey Inst of TechnologyDepartment of Mathematical [email protected]

MS6

Stability and Variability of Central Pattern Gener-ation

What is the target activity of homeostatic regulation ofnetwork performance? The pyloric rhythm of the lob-ster stomatogastric ganglion shows substantial animal-to-animal variability in frequency but the phase relationshipsbetween different neurons within the circuit are relativelytightly constrained. Individual networks with similar meanmotor output can behave differently on a cycle-to-cycle ba-sis, and model networks show that different combinationsof intrinsic and synaptic conductances can yield similar ac-tivity.

Dirk BucherBrandeis UniversityVolen Center for Complex [email protected]

106 DS05 Abstracts

MS6

Regulation of Bursting Activity in a NeuronalModel

Bursting activity of neurons is an oscillatory activity con-sisting of intervals of repetitive spiking separated by in-tervals of quiescence. Neurons which are capable of gen-erating bursting activity endogenously play an importantrole in Central Pattern Generators. Burst duration, in-terburst interval and spike frequency are crucial temporalcharacteristics of bursting activity and thus have to be reg-ulated.Application of the bifurcation theory of dynamicalsystems suggests new mechanisms of how bursting activitycan be generated by neurons and its temporal characteris-tics can be regulated. The work is supported by NIH NS43098.

Gennady S. CymbalyukGeorgia State UniversityDepartment of Physics and [email protected]

Andrey ShilnikovGeorgia State UniversityDepartment of Mathematics and [email protected]

MS6

Electrically Coupling Distinct Neurons

We developed a multi-compartment model of two intrinsi-cally distinct, electrically coupled neurons, inspired by thepacemaker group of the crustacean pyloric network. Weuse the model to illustrate several neuron and network be-haviors that arise from electrically coupling very differentoscillators. We explore conditions under which a small,intrinsically bursting neuron can drive a large, tonic spik-ing neuron to burst in-phase with it, and show that currentsegregation significantly enhances their ability to burst syn-chronously.

Eve MarderBrandeis [email protected]

Farzan NadimNew Jersey Institute of [email protected]

Pascale RabbahRutgers University at [email protected]

Cristina Soto-TrevinoNew Jersey Institute of TechnologyDepartment of Mathematical [email protected]

MS7

Instability of Winds over Mountainous Terrain

In a density-stratified atmosphere, winds over mountainsgenerate waves due to oscillations of vertical motion andbuoyancy. These wave disturbances are a significant contri-bution to the variations of cloudiness and precipitation inalpine regions. They are also an important loss mechanismin the energy and momentum budgets for the global cli-mate. R. Long (1953) made the remarkable discovery that,

in a special case of two-dimensional stratified flow, the non-linear steady streamfunction satisfies the linear Helmholtzequation. Surprisingly, it is only recently that very care-ful flow simulations for a series of two or three mountainsfound that Long’s solutions could be unstable. A numeri-cal linear stability analysis, which uses an Arnoldi approachfor the large sparse finite-difference system, produces un-stable eigenmodes which compare well with the simulatedinstability in the hydrostatic parameter regime. A spectralanalysis of the eigenmodes suggests that triad resonancesplay a key role in the dynamics of these flows.

Youngsuk LeeMathematics Department, Simon Fraser UniversityBurnaby, BC, [email protected]

MS7

Effects of Stratification on the Variability of theDouble-gyre Wind-driven Ocean Circulation

The double gyre wind driven ocean circulation exhibits arich variety of behaviors, including variability on timescalesrelevant to climate dynamics. While much is known aboutthe role forcing and dissipation have in determining thevariability of the flow, the effects of stratification have beenless thoroughly studied. In the two layer model, there aretwo stratification parameters, which can be interpreted asthe depth and strength of the thermocline. Both of theseparameters independently affect the variability of the flow;increasing the strength or decreasing the depth of the ther-mocline each lead to chaotic variability, but the routes tochaos are distinct.

Cavendish McKayDepartment of MathematicsUniversity of [email protected]

MS7

Floquet Instability & Triad Resonance in a Strati-fied Fluid

An important fluid mechanical property of the Earth’s at-mosphere, is that its density is stably-stratified. Unlikethe case of unstable stratification (light fluid under heavy)where convective motions can arise, a stably-stratified fluidpermits buoyancy oscillations known as gravity waves.These motions result from localized vertical disturbancesof heavier air upward (or lighter air downward) which arethen subject to the restoring force of gravity. Such wavesare commonly experienced as in-fight turbulence when anaircraft flies through oscillatory fields of vertically movingair. In the absence of viscosity, the stratified fluid equationsadmit an exact, nonlinear plane wave solution. Although itwas established in 1976 that these finite-amplitude wavesare parametrically unstable, there is much about the non-linearity of this system which is poorly understood. A sta-bility analysis of this wave reveals a simple connection be-tween the Floquet eigenvalues of the linearized problemwith the resonant traces of triad interaction theory.

David J. MurakiDepartment of MathematicsSimon Fraser [email protected]

DS05 Abstracts 107

MS8

High Efficiency Mixing using the Shear Super-posiyion Micromixer

We present new experimental results showing fast and highefficiency mixing procedures for an active shear superpo-sition micromixer. The mixing process is studied numer-ically and experimentally using flow visualizations tech-niques. The numerical simulations are performed for 3-Dflows using Fluent. We quantify, numerically and exper-imentally, the degree of mixing achived using the MixingVariance Coefficient. We show that single channel mixingcan be very good when the amplitude and frequency arechosen carefully.

Igor Mezic, Frederic BottausciUniversity of California, Santa [email protected], [email protected]

MS8

Chaotic Advection in Three-Dimensional UnsteadyIncompressible Laminar Flow

We review results regarding an experimentally realizablethree-dimensional time-dependent nonturbulent fluid flowthat displays the phenomenon of global diffusion of passive-scalar particles at arbitrarily small values of the nonin-tegrable perturbation. This type of chaotic advection,termed resonance-induced dispersion, is generic for a largeclass of flows.

Julyan CartwrightLaboratorio de Estudios CristalograficosUniversity of [email protected]

MS8

Designing chaotic microflows in spherical volumes

Achieving thorough chaotic mixing inside small sphericalvolumes, such as liquid microdroplets suspended in a liquidsubstrate, has proved to be quite challenging due to the in-herently high symmetry of the problem. This talk will pro-vide an overview of the important results in this area anddescribe some recent theoretical developments and exper-imental approaches to manipulating the microflow insideliquid droplets.

Michael SchatzCenter for Nonlinear Science & School of PhysicsGeorgia Institute of [email protected]

Roman GrigorievGeorgia Institute of TechnologyCenter for Nonlinear Science & School of [email protected]

Vivek SharmaGeorgia Institute of [email protected]

MS8

Right Hand - Left Hand

Experimental work in developmental biology has recentlyshown in mice that fluid flow driven by rotating cilia in the

early stages of growth of vertebrate embryos is responsiblefor determining the left-right axis, with the heart on theleft of the body, the liver on the right, and so on. The roleof physics, in particular, of fluid dynamics, in the process isone of the important questions that remain to be answered.

Idan TuvalInstituto Mediterraneo de Estudios AvanzadosCSIC - Universidad de las Islas Baleares, [email protected]

MS9

Almost Invariant Sets for Complex Systems

Over the last years set-oriented numerical methods havebeen developed for the approximation of statistical char-acteristics of dynamical systems. These methods allow toapproximate not just invariant measures but also almost in-variant sets. These subsets define a macroscopic structurein state space for the underlying dynamical process. Herewe discuss the most recent numerical approaches for identi-fying almost invariant sets. The algorithms are illustratedby several challenging examples such as the approximationof chemical conformations for molecules.

Mirko Hessel-von MoloInstitute for MathematicsUniversity of [email protected]

Robert PreisInstitute for [email protected]

Michael DellnitzInstitute for MathematicsUniversity of [email protected]

MS9

Air Force Challenges in Networked Dynamical Sys-tems

Sharon HeiseAir Force Office Of Scientific [email protected]

MS9

Modeling, Analysis, and Design of Large Dynami-cal Systems in Industry

Clas JacobsonUnited TechnologiesResearch [email protected]

MS9

Representation of Dynamical Systems: Applicationto Building Systems

The Frobenius-Perron (F-P) operator construction forgraph partitioning will be discussed, with applications tomodel reduction and dynamic analysis of the Building sys-tems models. We describe mathematical problems associ-ated with meeting objectives for efficiency, security, peopleegress, and network control solutions in integrated building

108 DS05 Abstracts

systems design. We model these problems as convection-diffusion equations on a (building) graph. While the origi-nal graph is complex, we use the F-P operator constructionalong with coarse spatio-temporal scales for any controlobjective to obtain effective (low-dimensional) representa-tion of dynamics on the graph. Additionally, application ofgroup representation theory to take advantage of symme-tries in the problem, and hierarchical model developmentto obtain hierarchical control solutions will be discussed.We illustrate the approach by means of examples drawnfrom efficiency and security problems in buildings.

Prashant G. MehtaUnited Technologies Research [email protected]

MS10

Symbolic Dynamics in Stochastic Dynamical Sys-tems: A Way to Define Their Markov Partitions

Symbolic dynamics are an important method for topolog-ical description of deterministic dynamical systems, butthey rely on well choosing a generating partition. In par-ticular, a Markov partition allows for a sophic shift of fi-nite type. Despite the ubiquity of at least random per-turbations, there is much less work to allow symbolic dy-namical description of a stochastic dynamical systems. Wegeneralize the definitions of Markov partitions, through aGalerkins method approach, to allow for stochastic sys-tems.


Karol ZyckowskiInstytut Fizyki im. SmoluchowskiegoUniwersytet Jagiello’nski, [email protected]

MS10

Unstable Recurrent Patterns inKuramoto-Sivashinsky Dynamics

We test the “recurrent patterns” description of turbulenceon a Kuramoto-Sivashinsky model, deploying a new vari-ational method that yields a large number of numericalunstable spatiotemporally periodic solutions. For a smallbut turbulent system, the attracting set appears surpris-ingly thin. Its backbone are several Smale horseshoe re-pellers, well approximated by local return maps, each withgood symbolic dynamics. Global dynamics appears decom-posable into chaotic dynamics within such local repellers,interspersed by infrequent transitions between them.

Predrag CvitanovicDepartment of Physics and Center for Nonlinear ScienceGeorgia Institute of [email protected]

Yuehng LanPhysics, Georgia [email protected]

MS10

Estimating Entropy Rates with Bayesian Confi-

dence Intervals

Matthew KennelInstitute For Nonlinear ScienceUniversity of California, San Diego,[email protected]

MS10

Symbolic Dynamics of Coupled Map Lattices

We show that coupled lattices of unimodal maps admit asimple generating partition which can be analyzed usingthe methods of symbolic dynamics. These methods havebeen very successful at rigorously defining and classifyingchaotic motion in one and two dimensional maps. Cou-pled Map Lattices (CMLs) are the first known example ofa high-dimensional system that can be treated this way.Much like the one-dimensional case, we find a generalizedGray ordering, a set of maximal sequences, and rules fordetermining the admissibility of symbol sequences. A spe-cific case utilizing coupled tent maps is presented.

Shawn PethelU.S. Army [email protected]

MS11

Symmetry in Circulant Multiple Agent Systems

We study the dynamics of a collection of point-mass agentsevolving under linear pursuit control laws. It is shown thata sufficient condition for the planar symmetry groups Cm

and Dm to be invariant under the agent dynamics is thatthe closed-loop dynamics be circulant. Necessary condi-tions on the agent dynamics are also obtained. Next weexamine the stability of the finite symmetry groups in theplane under circulant dynamics, extending prior work byRichardson [T. Richardson. Stable polygons of cyclic pur-suit. Annals of Mathematics and Artificial Intelligence.vol. 31, pp. 147-172, 2001] and Bruckstein et. al. [A.M. Bruckstein, G. Sapiro, and D. Shaked. Evolutions ofplanar polygons, International Journal of Pattern Recogni-tion and Artificial Intelligence, vol. 9, no. 6, pp. 991-1014,1995] on the stability of the cyclic group Cm for agents in“cyclic pursuit”. Some other aspects such as determinationof collisions are explored.

Joshua MarshallUniversity of [email protected]

Mireille E. BrouckeUniversity of TorontoDepartment of Electrical Comp. [email protected]

MS11

Analysis and Design Tools for Distributed MotionCoordination

This talk presents novel mathematical tools useful to studythe motion of mobile autonomous agents. First, motioncoordination tasks are encoded into aggregate cost func-tions from Geometric Optimization. Second, the limitedcommunication between agents is modeled via proximitygraphs from Computational Geometry. Third, algorithmscorrectness is established via LaSalle invariance principlesfor non-deterministic systems in discrete and continuous

DS05 Abstracts 109

time. Finally, these tools are applied in motion coordina-tion examples such as deployment, rendezvous, and flock-ing.

Francesco BulloMechanical & Environmental EngineeringUniversity of California at Santa [email protected]

Sonia MartinezUniversity of California Santa [email protected]

Jorge CortesUniversity of California Santa [email protected]

MS11

From Nonsmooth Analysis and Geometric Opti-mization to Distributed Coordination Algorithms

In this talk, we discuss dynamical systems that solve disk-covering, sphere-packing and maximum-visibility prob-lems. We present aggregate cost functions from geomet-ric optimization that encode various notions of networkdeployment. We design and analyze a collection of dis-tributed control laws that are related to nonsmooth gradi-ent systems. The resulting dynamical systems promise tobe of use in robotic coordination problems for networkedrobots; in this setting, the distributed control laws cor-respond to remarkably simple local interactions betweenthe robots. We formally discuss to what extent the pro-posed coordination algorithms are spatially distributed.The technical approach relies on concepts from computa-tional geometry, nonsmooth analysis, and the dynamicalsystem approach to algorithms.

Francesco BulloMechanical & Environmental EngineeringUniversity of California at Santa [email protected]

Jorge CortesDepartment of Applied Mathematics and StatisticsUniversity of California, Santa [email protected]

MS11

Asynchronous Coordination of Multi-Agent Sys-tems

This paper is concerned with the coordination of a groupof n > 1 mobile autonomous agents which all move inthe plane with the same speed but with different head-ings. Each agent changes its heading from time to timeto a new value equal to the average of its present headingand the headings of its current “neighbors”. Although allagents use the same rule, individual headings are updatedasynchronously. By appealing to the concept of “analyticsynchronization”, it is shown that under mild connectivityassumptions of the underlying directed graph character-izing neighbor relationships, the local update rules underconsideration can cause all agents to eventually move in thesame direction despite the absence of centralized coordina-tion and despite the fact that each agent’s set of neighborschange with time as the system evolves.

Stephen MorseDepartment of Electrical Engineering

Yale [email protected]

Brian Anderson, Ming CaoYale Universitytba, [email protected]

MS12

Manipulation of Self Aggregation Patterns andWaves in a Reaction-Diffusion-System by OptimalBoundary Control Strategies

We show numerically how spatiotemporal behaviour likepattern formation in a two component nonlinear reactiondiffusion model of bacterial chemotaxis, described by a sys-tem of two coupled quasilinear PDEs, can be externallycontrolled. We formulate the control goal as an objectivefunctional and apply numerical optimization for the solu-tion of the resulting control problem. Due to model insuffi-ciencies and measurement noise feedback control strategiescan be applied to feed model answers in real-time backto the optimization process. Nonlinear model predictivecontrol (NMPC) for partial differential equations (PDE)is deemed to be crucial for feedback control of distributedparameter systems. We demonstrate the application of anefficient nonlinear model predictive control (NMPC) algo-rithm.

Ulrich Brandt-PollmannInterdisciplinary Center for [email protected]

Dirk LebiedzInterdisciplinary Center for Scientific ComputingUniversity of Heidelberg, [email protected]

MS12

Localized Feedback Control of Pattern FormingSystems: Successes and Failures

This talk will concentrate on simple pattern form-ing systems described by one-dimensional partial differ-ential equations such as Ginzburg-Landau, Kuramoto-Sivashinsky or Swift-Hohenberg equation. I will show howpattern formation in these systems can be suppressed (oranother pattern imposed) using feedback applied at one orboth boundaries. I will also show that such control breaksdown when the size of the system increases due to expo-nentially growing transient amplification of noise.


Andreas HandelEmory [email protected]

MS12

Feedback Control of Morphological Instabilities

Feedback control of morphological instabilities of crystal-lization fronts in directional solidification is considered: (i)global control of the long-wave monotonic instability in sys-tems with small segregation coefficient and (ii) local con-

110 DS05 Abstracts

trol of the short-wave oscillatory instability in systems withkinetic undercooling. In case (i) it is shown that globalfeedback control can prevent the finite-time blow-up andlead to the formation of localized structures. In case (ii) itis found that the local feedback control can suppress theinstability but the control efficiency depends on the delay.

Vadim PanfilovUniversity of Nevada at [email protected]

Alexander NepomnyashchyTechnionIsrael Institute of [email protected]

Alexander A. GolovinDepartment of Engineering Sciences and AppliedMathematicsNorthwestern [email protected]

Tatiana SavinNorthwestern [email protected]

Valeria GubarevaDepartment of MathematicsTechnion - Israel Institute of [email protected]

MS12

Controlling Synchronization in Oscillator Ensem-bles

We consider a possibility to suppress synchronous collec-tive dynamics in ensembles of globally coupled oscillators.We show that addition of a delayed feedback of the meanfield allows us to stabilize the asynchronous state irrespec-tively on the particular way of the influence. Theory forthe Kuramoto model is presented. We also present nu-merical modelling of neural ensembles and discuss possibleapplications in neuroscience.

Arkady Pikovsky, Michael RosenblumDepartment of PhysicsUniversity of Potsdam, [email protected], mros(-)agnld.uni-potsdam.de

MS12

Optical Manipulation and Control of Micro-ScaleFluid Flow

We describe experiments using an all-optical approach tocontrolling microscale fluid flow. Surface-tension gradi-ents are induced in a fluid by the thermal absorption oflight. The gradients can drive flow on unstructured liquidor solid substrates; moreover the flow can be altered andcontrolled by spatially and temporally varying the lightintensity. Optical control permits the flow to be dynami-cally reprogrammed, suggesting an approach for construct-ing fluidic devices analogous to microcomputing’s CPU.


Michael F. Schatz, Jennifer RieserCenter for Nonlinear Science and School of PhysicsGeorgia Institute of [email protected], [email protected]

MS13

Stability of Pulses in Mixed Type Equations

Christopher JonesDepartment of MathematicsUNC, Chapel [email protected]

MS13

Nucleation of localised patterns in the 2D Swift-Hohenberg equation

We study the nucleation of localised patterns in 2-dimensions using a variety of rigorous, formal and numer-ical techniques. We show that in a generic pattern for-mation equation, namely the Swift-Hohenberg equation,localised patterns are formed by a heteroclinic connectionfrom the trivial state to the cellular pattern. We presenta priori estimates on the attractor and direct variationalmethods are used to show rigorous existence of small am-plitude pulses in 2-dimensions. Using asymptotics we findan estimate for the heteroclinic connection to form in thecase that the cellular pattern consists of squares. Finally,we use numerical methods to explore the bifurcation se-quence of the nucleation of localised patterns.

David Lloyd, Alan ChampneysUniversity of [email protected], [email protected]

MS13

A Mountain Pass and Gradient Flow for Stable So-lutions in Cylinder Buckling

A classical problem in structural engineering is the predic-tion of the load-carrying capacity of an axially compressedcylindrical shell. We obtain a single-dimple solution as amountain-pass point which is unstable, in the sense thatthere are directions in state space in which the energy de-creases. In one direction the dimple roughly shrinks anddisappears, and in the other direction it grows and multi-plies into a periodic array of dimples. This gives insightinto scaling properties and prediction of failure loads.

Jiri HorakUniversit”at K”[email protected]

Mark PeletierTechnische Universiteit [email protected]

Gabriel J. LordHeriot-Watt [email protected]

MS13

Numerical Evaluation of the Evans Function by

DS05 Abstracts 111

Magnus Integration

We use Magnus methods to compute the Evans function forspectral problems as arise when determining the linear sta-bility of travelling wave solutions to reaction-diffusion andrelated partial differential equations. In a typical applica-tion scenario, we need to repeatedly sample the solutionto a system of linear non-autonomous ordinary differentialequations for different values of one or more parametersas we detect and locate the zeros of the Evans functionin the right half of the complex plane. In this situation,a substantial portion of the computational effort—the nu-merical evaluation of the iterated integrals which appear inthe Magnus series—can be performed independent of theparameters and hence needs to be done only once. Moreimportantly, for any given tolerance Magnus integratorspossess lower bounds on the step size which are uniformacross large regions of parameter space and which can beestimated a priori. We demonstrate, analytically as wellas through numerical experiment, that these features ren-der Magnus integrators extremely robust and, dependingon the regime of interest, efficient in comparison with stan-dard ODE solvers. Authors: NAIRO D. APARICIO, SI-MON J.A. MALHAM and MARCEL OLIVER

Simon MalhamHerriott-Watt UniversityDepartment of [email protected]

MS13

Blowing-Up Exact Solutions of Long-Wave Unsta-ble Thin Film Equations

Long-wave unstable thin film equations

ht = (hnhxxx)x −B(hmhx)x

are a fourth-order analogue of the the semilinear heat equa-tion. A ”reaction” term destabilizes a ”diffusion” term, al-lowing for a competition between effects. This competitionadmits a variety of steady states and temporal behaviors,depending on whether the equation is subcritical, critical,or supercritical. In the critical case, Witelski, Bertozzi,and Bernoff propose a critical mass, Mc, and demonstratethat if the mass of the initial data is less than Mc thenfinite-time blow-up is impossible. Their computations andasymptotics case suggest that (generically) if the initialmass is larger than Mc the resulting solution demonstratesselfsimilar blowup focussed at points. Slepcev and I con-sider the critical case and construct exact solutions withcompact support that blow up in a selfsimilar manner.Their mass can be arbitrarily close to the critical massproposed by Witelski et al., proving the sharpness of thecritical mass. In addition, Slepcev has proven the linearstability/instability of such solutions.

Dejan SlepcevUCLAMathematics [email protected]

Mary PughUniversity of TorontoDepartment of [email protected]

MS14

Braid-Theoretic Methods for Parabolic PDEs

Robert W. GhristDepartment of MathematicsUniversity of Illinois, [email protected]

MS14

Classification of Strange Attractors in Three Di-mensions

It is finally possible to classify low-dimensional strange at-tractors. There are four levels of structure in this classi-fication: (1) basis sets of orbits; (2) branched manifolds;(3) bounding tori; and (4) embeddings into R3. All fourlevels involve links of knots in very powerful ways. We de-scribe how singularities form the backbone of stretchingand squeezing processes that generate chaotic behavior.We conclude with a brief description of all the covers ofa universal image dynamical system, the horseshoe.

Robert GilmorePhysics Dept., Drexel Univ.Philadelphia, [email protected]

MS14

Topological Analysis of Experimental Data : KnotHolders and Topological Entropies

Knots and links formed by unstable periodic orbits in astrange attractor carry signatures of the stretching andsqueezing mechanisms that organize it. We review how thisproperty has been harnessed to design a robust method foranalyzing experimental chaotic data. We also report recentexperiments in a nonstationary system where the knot typeof an orbit has been used as an unambiguous signature ofchaos. Possible extensions to higher dimensions will alsobe discussed.

Marc LefrancPhLAM/Universite Lille I,[email protected]

MS14

Analysis of Low Dimensional Dynamics: Algo-rithms Based on the Conley Index

The Conley index theory allows one to conclude the exis-tence of a variety of invariant sets including fixed points,periodic, heteroclinic and homoclinic orbits, and symbolicdynamics. However, to apply the theory requires the con-struction of an isolating neighborhood and the computa-tion of the associated index. I will discuss algrorithms thatperform these constructions and computations.


MS14

Perestroikas of Strange Attractors in Three-

112 DS05 Abstracts

Dimensional Phase Space

Strange attractors can exhibit bifurcations just as periodicorbits in these attractors can exhibit bifurcations. We de-scribe two classes of large-scale bifurcations that strangeattractors can undergo. For each we provide a mechanism.These bifurcations are illustrated in a simple class of three-dimensional dynamical systems.

Tsvetelin D. TsankovPhysics DepartmentBryn Mawr College, Bryn Mawr, [email protected]

MS15

Stability of a Stripe in the Two-Dimensional Gray-Scott Model

We analyse the stability of a stripe solution of the Gray-Scott model in the weakly-coupled regime. We studythree different types of instabilities: a splitting instability,whereby a stripe self-replicates into two parallel stripes; abreakup instability, where a stripe breaks up into spots;and a zigzag instability, whereby a stripe develops a wavyperturbation in the transversal direction. We derive ex-plicit thresholds for all three types of instability. Someopen problems will be discussed. This is a joint work withMichael J. Ward and Juncheng Wei.

Juncheng WeiDepartment of MathematicsChinese University of Hong [email protected]

Theodore KolokolnikovFree University of [email protected]

Michael WardDepartment of MathematicsUniversity of British [email protected]

MS15

Can Weak Interaction Cause Annihilation?

We discuss about annihilation-repulsion criterion for thescattering dynamics of particle-like patterns in dissipativesystems. Hidden saddles, called ”scattors”, essentially con-trol the input-output relation during the scattering process.For non-fusion type of scattering, a scattor with codim 3singularity consisting of saddle-node, Hopf and drift insta-bilities is responsible for the onset of annihilation-repulsionboundary in a parameter space. This implies that annihi-lation occurs for arbitrary small velocity of traveling pat-terns.

Yasumasa NishiuraRIES, Hokkaido [email protected]

Kei-Ichi UedaRIMS, Kyoto [email protected]

Takashi TeramotoChitose Institute of Science and [email protected]

MS15

Semi-Strong Pulse Interaction

Pulse-pulse interactions play central roles in a variety ofpattern formation phenomena, including self-replication.From the analytical point of view, pulse interactions can bedistinguished into three types: weak interactions, in whichthe pulses are assumed to be sufficiently far apart, the fullystrong interactions, and the intermediate concept of semi-strong interactions, that has been introduced in the contextof singularly perturbed systems, in which only some com-ponents of the pulse interact weakly. Recently, methodsbased on constructing (approximate) invariant manifoldsand/or renormalization techniques have been developed torigorously study weak pulse interactions. In this talk thesemethods will be extended, so that they can be applied tosemi-strong pulse interactions. This talk is based on jointwork with Arjen Doelman (Amsterdam) and Keith Promis-low (Michigan).

Arjen DoelmanCWI Amsterdam, the [email protected]

Tasso J. KaperBoston UniversityDepartment of [email protected]

Keith PromislowMichigan State [email protected]

MS15

Pulse Splitting on Growing Domains

Pattern formation on growing domains has attracted in-terest in the past few years due to its biological applica-tions, e.g. for the patterning of animal skins. A recurrentphenomenon is the repeated splitting of patterns duringgrowth, in particular for Turing patterns. We present fur-ther analysis of this, also for excitation pulses, in terms ofstability and bifurcation considerations with focus on theonset of splitting. This is joint work with Michael Ward.


Jens RademacherUniversity of British ColumbiaDepartment of [email protected]

MS15

Competition and Oscillatory Instabilities of Spikesin the Semi-Strong Interaction Limit

The dynamics and instability mechanisms of one-spike andtwo-spike solutions to the Gierer-Meinhardt and Gray-Scott models are analyzed on a bounded one-dimensionaldomain. For each of these non-variational two-componentsystems, the semi-strong spike-interaction limit, where theratio of the two diffusion coefficients is asymptoticallylarge, is analyzed. In this limit, differential equations forthe slow time-dependent motion of the locations of thespikes are derived. The stability of these solutions on a fasttime scale is studied by a spectral analysis of certain non-

DS05 Abstracts 113

local eigenvalue problems that involve the instantaneousspike locations. From these nonlocal eigenvalue problemsit is shown that eigenvalues can enter into the unstableright half-plane along either the real axis or through a Hopfbifurcation, leading to either a competition instability ora synchronous oscillatory instability of the spike pattern.The important feature of these instabilities are that theycan be dynamic in nature, and so can be triggered at somepoint during the slow evolution of a spike pattern thatis initially stable at time t=0. The asymptotic theory iscompared with full numerical results. Competition andsynchronous oscillatory instabilities are also shown to oc-cur for k-spike equilibria in a one-dimensional domain, andin a two-dimensional spatial domain.




Wentao SunDepartment of MathematicsSimon Fraser [email protected]

MS15

Stripes and Wriggled Stripes in Reaction-DiffusionSystems

We consider the existence and stability of perfect and wrig-gled stripes for two reaction-diffusion systems: First one isGierer-Meinhardt system with saturation:

at = ε2∆a− a+a2

h(1 + ka2)

τht = D∆h− h+ a2

where k > 0 is a fixed paramter; The second one is di-blockmodel

ε2∆u+ u− u3 + γε(−∆)−1(u− a) = Constant

Both equations are considered in a perfect rectangle R =[0, 1]× [0, 1]. The stability of N− stripes is completely de-termined by the small eigenvalues which we have computedexplicitly. Our results show that the stripes can be stablein some cases.


MS16

Reduced Model for Twist Induced Instabilities ofScroll Waves

We present a reduced model for the dynamics of the meanfilament and twist of a scroll wave. Using the model, as well

as direct numerical simulations of reaction diffusion sys-tems, we study several aspects of the dynamics of twistedscroll waves: the propagation of pulses appearing from asecondary instability, the build-up of twist in systems withspatially varying excitability, and the effect of conductionanisotropy on the formation of twist.

Vincent HakimLPS, Ecole Normale [email protected]

Blas EchebarriaUniversitat Politecnica de [email protected]

MS16

Contributions of Cellular and Structural Hetero-geneity to Rabbit Ventricular Arrhythmia

Analysis of cardiac electrophysiology in intact myocar-dial tissue is made more complex by the presence of cel-lular and structural heterogeneities, as well as a num-ber of interacting biophysical processes, including neuro-hormonal control. We present here computational mod-els that integrate detailed cellular systems models of car-diomyocyte excitation-contraction coupling and its regu-lation into anatomically detailed 3D continuum models ofthe rabbit ventricles. These models may prove useful toolsin elucidating the mechanisms of arrhythmogenesis.

Andrew McCullochUniversity of California, San DiegoDepartment of [email protected]

Sarah HealyUniversity of California San DiegoDepartment of [email protected]

MS16

A Tridomain Model of Discrete 3D Cardiac Tissue

Most models used to study the cardiac electrodynamics as-sume the tissue structure as being either a continuous mon-odomain or a bidomain. For many diseased states, how-ever, the cell coupling is heterogeneous, impacting wave-front propagation. We have recently developed a tridomainmodel of cardiac tissue in which each cell is a discrete threedimensional object, enclosed by a membrane and embed-ded in a discrete interstitial domain. We present simulationstudies comparing the model with traditional models.

Sarah Roberts, Jeroen Stinstra, John PormannDuke [email protected], [email protected], [email protected]

Craig HenriquezDept of Biomedical EngineeringDuke [email protected]

MS16

Computational Infrastructure for Biomedical Mod-eling, Simulations, and Visualizations

Computational infrastructures provide the framework inwhich computing can support a particular application. A

114 DS05 Abstracts

good infrastructure can be critical to a project as it de-termines how a program (or set of programs) looks, feels,and is amenable to extension and change. Problem solvingenvironments (PSEs) are a form of computational infras-tructure that seeks to provide an integrated set of tools fora particular application area. In this talk, I discuss our re-cent research and development of component-based PSEsfor biomedical computing.

Chris JohnsonUniversity of UtahDepartment of Computer [email protected]

MS16

Ventricular Fibrillation Activation Patterns on theSurface of the Swine Heart

We recorded ventricular fibrillation (VF) activation pat-terns from the surface of Langendorff-perfused swine heartsusing a new panoramic optical mapping system that allowsalmost all of the epicardium to be mapped with 1-2 mmresolution at 750 fps. Individual phase singularities didnot persist for more than about 500 ms; however, unin-terrupted epicardial wavefronts could be tracked for entiremapped episodes (4 s). This indicates that VF mainte-nance can be explained by multiple-wavelet epicardial reen-try.

Matthew Kay, James GladdenDepartment of Biomedical EngineeringUniversity of Alabama at [email protected], [email protected]

Gregory WalcottDepartment of MedicineUniversity of Alabama at [email protected]

Jack RogersUniversity of Alabama at [email protected]

MS16

Complex Electrical Dynamics in a Simulated HeartSlice

Cardiac tissue engineering has opened up the possibilityof studying electrical wave-front dynamics in complex butcontrolled substrates. An in silico method is presented thatmimics the structural development of these cultures fromprotein patterning, through cell spreading to intercellularcoupling. The electrical behavior of these in silico will helpdesign experimental studies and interpret results.

Joseph TranquilloDuke [email protected]

MS17

Multiscale Modeling of Biological Systems

The mechanics associated with many biological processescan be modeled to a first approximation by elastic struc-tures which interact with a fluid. At small length scalesthermal fluctuations play a significant role in system dy-namics. In this talk we discuss how to extend the im-mersed boundary method of Peskin using stochastic forc-

ing to model thermal fluctuations. Developing numericalmethods which capture the dynamics of the system wellis made difficult by the range of time scales associatedwith the degrees of freedom of the fluid and elastic struc-tures. We show how a numerical method which achieveslong time steps can be developed by approximating thefastest degrees of freedom of the system using an appropri-ate stochastic model. A statistical analysis of the behaviorof particles, polymers, and membrane-like sheets will bepresented for the method. We will further discuss applica-tions of the method to biological and microfluidic systems.

Peter R. KramerRensselaer Polytechnic InstituteDepartment of Mathematical [email protected]

Paul AtzbergerRensselaer Polytechnic [email protected]

MS17

Nonlinear Proper Orthogonal Decomposition forMolecular Systems

We consider the method of proper orthogonal decomposi-tion (POD) on functions of Cartesian molecular data, e.g.,on torsion angles. By pulling back the nonlinear POD in-formation thus obtained to the original model equations wecan decompose the system in a nontrivial way, and derivea reduced model. If we moreover replace truncated modesby an appropriate stochastic process the reduced modelcaptures the essential features of the original dynamics.

Christof SchuetteUniversity of [email protected]


Carsten HartmannFree University BerlinInstitute of [email protected]

MS17

Stochastic Mode Reduction with Metastability inBiomolecular Modeling

One approach to accelerating biomolecular simulations isthrough a mode reduction in which only certain “slow” de-grees of freedom of interest are simulated, and the influenceof the remaining “fast” variables are incorporated througheffective deterministic and stochastic terms. Metastabilityis a prevalent feature in biomolecular systems, and we illus-trate, through detailed analysis of a simple model problem,various ways in which it can affect the effective equationsobtained from a mode reduction procedure.


Peter R. Kramer

DS05 Abstracts 115

Rensselaer Polytechnic InstituteDepartment of Mathematical [email protected]

Jessika Walter, Wilhelm HuisingaFree University of BerlinInstitute of [email protected], [email protected]

Carsten HartmannFree University BerlinInstitute of [email protected]

MS17

The Stochastic Spectral Dynamics of Bending andTumbling

Traditional models of wormlike chains in shear flows at fi-nite temperature approximate the equation of motion viafinite difference discretization (bead and rod models). Weintroduce here a new method based on a spectral represen-tation in terms of the natural eigenfunctions. This formu-lation separates tumbling and bending dynamics, clearlyshowing their interrelation, naturally orders the bendingdynamics according to the characteristic decay rate of itsmodes, and displays coupling among bending modes in ageneral flow. This hierarchy naturally yields a low dimen-sional stochastic dynamical system which is amenable toanalytic treatment (recovers and extends previous numer-ical results on stability of rods in shear) and which leadsto a fast and efficient numerical method for studying thestochastic nonlinear dynamics of semiflexible polymers ingeneral flows.

Chris WigginsDept. Applied Physics and MathematicsColumbia [email protected]

Alberto Montessi, Matteo PasqualiRice [email protected], [email protected]

MS18

Stability and Robustness of Channel Flows withCorrugated and Flexible Walls

We study wall bounded turbulent flows with either corru-gated or flexible walls. Numerical and experimental studieshave shown that both types of walls produce reductions inskin-friction drag. However, no systematic theory of thedynamical effects of such walls exist. We study the effectof such walls on robust stability and noise amplification ofthe underlying linearized Navier-Stokes equations.

Bassam A. BamiehUC Santa [email protected]

MS18

Fundamental Limitations in the Control of FluidSystems

A new area for the mathematical machinery for Navier-Stokes systems is in the characterization of the fundamen-tal performance and stabilization limitations in represen-tative flow control problems, such as the minimum drag

attainable with zero-net mass flux controls in a channelflow and the maximum Reynolds number for stabilizabil-ity of vortex shedding in the flow past a cylinder. This talkwill summarize the first result established of this kind forthe Navier-Stokes equation.

Thomas BewleyUniversity of California, San DiegoMechanical and [email protected].

MS18

Robust Stability and Control of Wall Bounded Tur-bulence

In this talk the limitations of traditional hydrodynamic sta-bility theory and control are shown and a framework for ro-bust flow stability and control is formulated. A host of newtechniques like gramians, singular values, operator norms,etc. are introduced to understand the role of various kindsof uncertainty. An interesting feature of this framework isthe close interplay between theory and computations. It isshown that a subset of Navier-Stokes equations are glob-ally, non-nonlinearly stable for all Reynolds number. Yet,invoking this new theory, it is shown that these equationsproduce structures and features as seen in our experiments.Feedback control results are presented.

John DoyleCaltechDepartment of Control and Dynamical [email protected]

Kumar M. BobbaUniversity of MassachusettsMechanical and Industrial [email protected]

MS18

Panel Discussion

The speakers will address some of the key unsolved prob-lems in theoretical and computational robust fluids sta-bility and control arena. The panel will also try to iden-tify some of our current limitations or “bottlenecks” fromthe mathematical and simulation side towards reaching theabove goals—in an effort to speculate what can be expectedin the coming years. The relevance of these developmentsto experimental fluid mechanics and technological applica-tions of fluid flow in aerospace and mechanical engineeringwill be touched upon.

Kumar M. BobbaUniversity of MassachusettsMechanical and Industrial [email protected]

MS18

Stability, Turbulence and Direct Numerical Simu-lation

Robert MoserUniversity of [email protected]

MS18

Control of Swept Attachment-Line Boundary Lay-

116 DS05 Abstracts

ers

Swept attachment-line boundary layers form at the lead-ing edge of any blunt body moving through a fluid at anon-zero yaw angle. Using an optimization scheme basedon a variational approach, we investigate disturbances fa-vored by the nonhomogeneous mean flow and assess theirtemporal evolution and growth mechanism. The same op-timization scheme is then used to apply an optimal blow-ing/suction strategy to minimize the rise of previouslyidentified instabilities.

Peter SchmidUniversity of Washington, SeattleDepartment of Applied [email protected]

MS19

Interplay Between Excitation and Inhibition Dur-ing Seizures

The dynamics of seizures in the CA1 region of the hip-pocampus is largely sculpted by the behavior of interneu-rons. Excessive inhibitory activity leads to a widespreadfailure of inhibition as these cells are transiently forced intodepolarization block. The loss of effective inhibitory actionreleases a significant excitatory response. The mechanismsfor these network interactions are probed through novelanalysis of experimental data along with simulations usingnetworks of conductance based neurons.

John R. Cressmankrasnow institutegeorge mason [email protected]

MS19

Controlling Wave Propagation in Cortex Theoryand Experiment

We experimentally confirmed predictions that modulationof neuronal threshold with electrical fields can speed up,slow down, and even block traveling waves in neocorticalslices. The predictions are based on a Wilson-Cowan typeintegrodifferential equation model of propagating neocor-tical activity. Wave propagation could be modified quicklyand reversibly. To the best of our knowledge, this is thefirst example of direct modulation of threshold to controlwave propagation in a neural system.

Bruce GluckmanGeorge Mason [email protected]

MS19

How Synaptic Dynamics May Organize Seizures

The dynamics of seizures involves the imbalance betweenexcitatory and inhibitory cell populations. Recent experi-ments have observed periodic interactions between distinctneuron populations. I will give a bifurcation analysis for abiophysical model for a two population network. The keydynamical mechanisms are a Hopf bifurcation due to dy-namical synapses, and periodic fast-slow dynamics due toan increase of ion concentration during elevated firing. Wealso give predictions for the event propagation.

Evelyn SanderGeorge Mason University

[email protected]

MS19

Spiral Waves in Mammalian Brain - Experiments

We report stable rotating spiral waves in mammalian tan-gential brain slices visualized by voltage-sensitive dye imag-ing. Spiral waves occurred spontaneously and alternatedwith plane, ring, and irregular waves. Spiral rotation rateswere about 10 turns per second, and the rotation was linkedto the oscillations in a one-cycle-one-rotation manner. Asmall slowly drifting phase singularity occurred at the cen-ter of the spirals. Such spiral waves may provide a spatialframework to organize cortical oscillations.

Steven J. SchiffGeorge Mason UniversityThe Krasnow [email protected]

Jian-Young WuGeorgetown UniversityPhysiology and [email protected]

Xiaoying HuangGeorgetown [email protected]

MS19

Modulating Neuronal Synchronization with Elec-trical Fields - Theory

Two heterogeneous model neurons were synaptically cou-pled and embedded within a resistive array, thus allowingthe neurons to interact both chemically and electrically.An applied electric field was found to be effective in con-trolling the transition of synchrony between these neurons.A simple phase oscillator reduction was successful in qual-itatively reproducing these results. These findings suggesta larger scale model in which the effects of electric fieldson seizure activity may be simulated.

Paul SoGeorge Mason UniversityThe Krasnow [email protected]

MS19

Spiral Waves in Brain Theory

We investigate wave formation in a two dimensionalWilson-Cowan type model of neural media. Two dimen-sional waves that are observed in the model include rotat-ing spirals, ring shaped plane waves, and periodic waves.All of these have experimental counterparts recently dis-covered in the occipital cortex of the rat. Movies will beshown which compare both model and experimental re-sults.

William TroyUniversity of PittsburghDepartment of [email protected]

MS20

Practicalities of Implementing a Bistatic Link Us-

DS05 Abstracts 117

ing Chaotic Signals

The practicalities of transmitting a continuous chaoticwaveform from a drifting sonar projector to a drifting re-ceiver are discussed. These include the trade-off betweentransducer characteristics, target properties and signalcharacteristics, the need to communicate a signal replicato the remote receiver for matched filtering and the effectof multipaths. Experimental results from a sea trial areused to illustrate the arguments.

P. R. AtkinsUniversity of BirminghamDept. of Electronic, Electrical and Computer [email protected]

MS20

Testing Chaotic Waveforms for Radar

Bandwidth increases distance resolution in a radar wave-form, but not all broad band waveforms are good for radar.We look at waveforms from several chaotic systems, includ-ing several Rossler like systems and a recently publishedhyperchaotic system. We plot ambiguity diagrams forthese waveforms, and compare to bandpass filtered noiseor a linear FM chirp. Finally, using results from a wave-form that was actually transmitted, we show that one mayidentify which one of several chaotic signals was transmit-ted.

Thomas CarrollUS Naval Research [email protected]

MS20

Bifurcation, Bursting, Detection and Classification

The use of chaotic signals provides opportunities for noveldetection and classification schemes in radar and sonar.In particular when there are non-linear interactions, bifur-cations may occur. Some situations will be given. Withlinear systems, bursting can occur in chaotic echoes underconditions which will be described. The use of bursting toprovide an indication of the presence or type of target willbe discussed.

Alan [email protected]

MS20

Parameter Selection of Discrete Chaotic Maps forImproved FM Ambiguity

We recently proposed the use of first-order chaotic maps toconstruct wideband FM signals for radar imaging. In par-ticular we showed that an FM signal based on the Bernoullimap exhibited a near optimum ambiguity surface. In sub-sequent analyses we have observed that the Bernoulli mapcan yield highly correlated samples. The degree of cor-relation depends on the choice of the B parameter of thechaotic map. We show that only specific values of B yielduncorrelated samples, which improve the characteristics ofthe chaotic FM signal’s ambiguity surface.

Benjamin C. FloresUniversity of Texas at El PasoElectrical and Computer Engineering

[email protected]

Gabriel ThomasElectrical and Computer [email protected]

Berenice VerdinElectrical and Computer EngineeringUniversity of Texas at El [email protected]

MS20

Chaotic Waveforms for Radar Applications

Kim ScheffUS Naval Research [email protected]

MS20

Robust Chaotic Signal Detection in DispersiveChannels

Radar, sonar and communication systems conventionallyuse coherent detection to improve reliability but this re-quires accurate synchronisation between transmitter andreceiver. Identical chaotic synchronisation has been pro-posed and demonstrated, but performance is poor in noisyand distorting channels. This presentation will discuss lessrestrictive definitions of synchronisation, and demonstratethe impact of typical radio and underwater channels onchaotic signals and synchronisation performance. Further,processing techniques to improve detection performancewill be analysed and discussed.

Christopher WilliamsUniversity of [email protected]

MS21

Delay Effects in Applications: A Quasi-HistoricalIntroduction

Time delays (or lags or retardations) have been recognizedfor quite some time to play a role in the dynamics of numer-ous regulatory systems, particularly those with feedback.As an introduction to this minisymposium, whose goal isto illustrate current knowledge of the analysis of delayedsystems in applications from a dynamical point of view,we present, with a historical perspective, examples fromdecades and centuries past to illustrate how the explicitincorporation of time delays in the modeling of biologicaland physical systems yields a more faithful mathematicalrepresentation. The importance of the modeling procedureitself will not be underestimated. In addition, and perhapsmore importantly, we also argue that contemporary tech-niques from dynamical systems theory can be incorporatedin the analysis of these models.

Jacques BelairUniversity of MontrealDepartment of [email protected]

MS21

Restrictions on Dynamics in Biological Delay Mod-

118 DS05 Abstracts

els at Double Hopf Bifurcation

This talk will be concerned with the restrictions on dy-namics at double Hopf bifurcations for a delay model of amotor control task experiment with Parkinsonian patientsand the delay model of the pupil light reflex. I will beshown that the restrictions on the dynamics are delay de-pendent and a consequence of the structure of the models.Further research directions will be discussed.

Jacques BelairUniversity of MontrealDepartment of [email protected]

Pietro-Luciano BuonoUniversity of Ontario Institute of TechnologyFaculty of [email protected]

MS21

Stability and Bifurcation Analysis of a NonlinearDelay Equation Model for Drilling

We study a delay differential equation model for chatter intwist drills in which a linear vibration mode interacts withnonlinear cutting forces. The model describes an oscilla-tor with nonlinear damping and cross-terms in the damp-ing and the delay. Linear stability analysis and nonlinearanalysis of the primary Hopf bifurcation, by constructinga centre manifold using symbolic algebra, shows that thestability of the Hopf bifurcation depends on the vibrationtype and the cutting speed.

Emily F. StoneThe University of MontanaDepartment of Mathematical [email protected]

Sue Ann CampbellUniversity of WaterlooDept of Applied [email protected]

MS21

Bifurcations of Mutually Delay-Coupled Lasers

Delay-coupled semiconductor lasers are an attractivechoice to study effects of time delay in coupled nonlin-ear oscillators. We present a bifurcation study of a delaydifferential equation model of two identical, delay-coupledsemiconductor lasers. We find a comprehensive geometri-cal picture that explains the bifurcations of solutions calledcompound laser modes in dependence on two physicallyrelevant parameters, namely the coupling phase and thedetuning.

Daan LenstraTheoretical PhysicsVrije Universiteit [email protected]


Hartmut ErzgraberVrije Universiteit Amsterdam

Afdeling Natuurkunde en [email protected]

MS21

Dynamics of Systems with Delayed Relay Control

A ubiquitous phenomenon in control is the so-called criti-cal delay. That is, if the delay in the control loop is largerthan a critical value, linear stabilization by feedback be-comes impossible. In contrast, a relay switch can stabilizean unstable system toward stable periodic motion even ifsubject to a large delay. The balanced inverted pendulumserves as illustrative example, also in a discussion of thedynamics of piecewise smooth systems with delay.

Jan SieberUniversity of BristolDept. of Eng. [email protected]

MS21

Wheel Shimmy Caused by Distributed Delays

A non-holonomic model of the shimmying wheel is stud-ied that uses the nonstationary values of the distributedcontact force system between the elastic tire and the rigidground. The governing equations are coupled partial andintegral differential equations. The existence of a travellingwave-like solution of the PDE leads to the RFDE

V 2ψ(t)+ψ(t) =L− 1

L2 + 1/3

∫ 0

−1

(L−1−2θ)ψ(t+θ)dθ+h.o.t.

where ψ is the caster angle, and the parameters V and Lare the dimensionless speed of towing and caster length,respectively.

Denes Takacs, Gabor StepanBudapest University of Technology and EconomicsDepartment of Applied [email protected], [email protected]

MS22

Synchronizability of Complex Networks

We explore the interplay of network topology and synchro-nizability in the classic Kuramoto model of coupled non-linear oscillators. We go beyond the existing results forall-to-all networks of identical oscillators by allowing fornetworks of arbitrary interconnection topology and oscilla-tors with uncertain natural frequencies. We use tools fromspectral graph theory and control theory to relate graphproperties to the critical coupling above which all the os-cillators synchronize. We further explain the behavior ofthe system as a function of the number of oscillators as itgrows to infinity.

Ali JadbabaieDepartment of Electrical and Systems EngineeringUniversity of [email protected]

Mauricio BarahonaImperial College [email protected]

DS05 Abstracts 119

MS22

Synchronization of Bursting Neurons: What Mat-ters in the Network Topology

We study the influence of coupling strength and networktopology on synchronization behavior in pulse-coupled net-works of bursting Hindmarsh-Rose neurons. Surprisingly,we find that the stability of the completely synchronousstate in such networks only depends on the number of sig-nals each neuron receives, independent from all other de-tails of the network topology. This is in contrast with lin-early coupled bursting neurons where complete synchronystrongly depends on the network structure and number ofcells. Through analysis and numerics, we show that theonset of synchrony in a network with any coupling topol-ogy admitting complete synchronization is ensured by onesingle condition.

Igor BelykhSwiss Federal Institute of Technology [email protected]

MS22

Synchronization is Enhanced in Weighted Net-works

We show that synchronizability is enhanced in weightednetworks where the coupling retains information on thestructure of the shortest pathlengths connecting the nodes,or on the age of the nodes. This issues are formally treatedin relation with variations in the eigenvalues of the corre-sponding asymmetric connecting matrices, and examplesare provided for phase synchronization and complete syn-chronization in a class of scale-free networks of Roessleroscillators.

Dong-Uk Hwang, Andreas AmannIstituto Nazionale di OtticaFlorence, [email protected], [email protected]

Stefano BoccalettiIstituto Nazionale di Ottica [email protected]

Mario ChavezIstituto Nazionale di Ottica AppplicataFlorence, [email protected]

MS22

On the Phase Reduction and Response Dynamicsof Neural Oscillator Populations

We undertake a probabilistic analysis of the responseof repetitively firing neural populations to simple pulse-like stimuli. Recalling and extending results from theliterature, we compute phase response curves (PRCs)valid near bifurcations to periodic firing for Hindmarsh-Rose, Hodgkin-Huxley, FitzHugh-Nagumo, and Morris-Lecar models. Phase density equations are then used toanalyze the role of the bifurcation, and the resulting PRC,in responses to stimuli. In particular, we explore the in-terplay among stimulus duration, baseline firing frequency,and population level response patterns.

Eric T. BrownCourant Institute for the Mathematical SciencesNew York University

[email protected]

Jeff MoehlisDept. of Mechanical and Environmental EngineeringUniversity of California – Santa [email protected]

Philip HolmesProg. in Applied and Comp. MathematicsPrinceton [email protected]

MS22

Foring Synchrony: Spatial Correlations in NoisyCoupled Neurons

Ensembles of neurons often receive spatially correlated in-puts from both their environment and distant brain re-gions. I will show how the ensemble statistics of couplednonlinear oscillators/neurons are sensitive to a correlatedstochastic forcing. This interplay between self-organizedand driven synchronization allows for an interesting cod-ing strategy where spatially distributed information can becoded with a temporal ensemble response.

Brent DoironNew York UniversityCourant [email protected]

MS22

Synchronization Mechanisms of Minimal Networksof the Parahippocampal Region

Experimental and theoretical studies have shown that os-cillatory activity at theta frequencies (8-12 Hz) can be gen-erated by networks having fast-firing (I) interneurons (in-hibitory) and either oriens lacunosum-moleculare (O-LM)interneurons (inhibitory) or stellate (S) cells (excitatory).O-LM and S have an hyperpolarization-activated (h-) cur-rent in addition to the standard Hodgkin-Huxely ones. Weuse a map approach to explain the generation of the rhyth-mic activity at theta frequencies and the special role playedby the h-current.

Dmitri PervouchineBoston UniversityBoston, [email protected]

Nancy J. KopellBoston UniversityDepartment of [email protected]

Horacio RotsteinBoston UniversityBoston, [email protected]

MS23

The Dynamics of Modulated Wave Trains

In this talk, we consider the dynamics of nonlinear wavetrains in reaction-diffusion equations. We prove that slowlyvarying modulations of wave trains are well approximatedby solutions of the Burgers equation over the natural timescale. In addition to the validity of Burgers equation over

120 DS05 Abstracts

long but finite time intervals, we show that the viscousshock profiles in Burgers equation can be found as genuinemodulated waves, or defects, in the underlying reaction-diffusion system. Moreover, we establish the stability ofthese defects. This talk is based on joint work with BjornSandstede, Arnd Scheel and Guido Schneider.


MS23

Asymptotic Validity for Discrete Equations Arisingin Nonlinear Optics

Solitons in discrete optical systems have been a topic ofkeen interest over the past decade. In a diffraction man-aged waveguide array, where the waveguide’s diffractionprofile is varied periodically in the direction of propagation,the slow propagation of the electric field envelope is mod-eled by a nonlocal discrete equation that exhibits solitonsolutions numerically. We verify the validity of the asymp-totic approximation, demonstrating that solutions of theaveraged equation are close in the l2 sense to those of theoriginal model for long time scales, and moreover, provethat the averaged equation has stable stationary solutions,complementing previous experimental and numerical ob-servation of these solutions.

Jamison T. MoeserUniversity of [email protected]

MS23

Justification of the Nonlinear Schrodinger Equa-tion in Spatially Periodic Media

The dynamics of the envelopes of spatially and temporar-ily oscillating wave packets advancing in spatially peri-odic media can approximately be described by solutionsof a Nonlinear Schrodinger equation. Here we prove es-timates for the error made by this formal approximationusing Bloch wave analysis, normal form transformations,and Gronwall’s inequality. Joint work with Kurt Busch,Guido Schneider, Lasha Tkeshelashvili

Hannes UeckerUniversity of KarlsruheDepartment of [email protected]

MS23

Approximations for the Collision of Two SolitaryWater Waves

The KdV approximation for surface water waves in a shal-low canal predicts both a head on collision of solitary wavesas well as an overtaking collision. While such collisions arein fact observed experimentally, there are notable depar-tures from the predictions of the KdV model. We discussthese departures and propose a set of equations which gov-ern corrections to the KdV approximation. These consistof linearized and inhomogeneous KdV equations plus an in-homogeneous wave equation. Some comparisons betweenthis higher order model and experimental data taken by J.Hammack and D. Henderson will be shown.

Doug WrightUniversity of Minnesota

School of [email protected]

MS24

Self-Averaging Scaling Limits of Waves in Turbu-lent Media

We show under fairly general assumptions on the turbulentrefractive index field that sufficient amount of medium di-versity leads to statistical stability or self-averaging of wavepropagation in the sense that its limiting law is determinis-tic and is governed by one of the 6 different types of trans-port (Boltzmann or Fokker-Planck) equations dependingon the specific scaling involved. We discuss the connectionto the statistical stability of time reversal procedure.

Albert FannjiangDepartment of MathematicsUniversity of California at [email protected]

MS24

Non-Standard Homogenization for an Advection-Diffusion Problem

We describe some interesting asymptotic features whichemerge in a simple model for turbulent diffusion consistingof a large scale mean flow and small scale periodic fluctu-ations. While standard homogenization theory describesone distinguished limit, the dynamics are more complexwhen the mean flow is strong. We investigate the effectivetransport through multiple scale analysis (involving threetime scales) and numerical simulations.

Peter R. Kramer, Adnan KhanRensselaer Polytechnic InstituteDepartment of Mathematical [email protected], [email protected]

MS24

Eddy Viscosity of Cellular Flows

In two dimensions in the presence of small-scale eddies thetransport of large-scale vector quantities may be accom-panied with depleted, and even “negative” diffusion. Thisphenomenon can be investigated by the stability analysisof the effective (averaged) equations. The eddy viscosityis a tensor in this equations. The main questions are thestructure of this tensor, how it depends on the underlyinggeometry of the small-scale eddies and how accurate thepredictions of the effective equations. The goal of this workis to answer these questions for a simple model where theeddies are given by a fixed periodic time-independent flow.A particular example is a cellular flow with the stream-function sin x/a sin y/a, a ¡¡ 1.

Alexei NovikovPenn State [email protected]

MS24

Periodic Homogenization for Inertial Particles

We study the problem of homogenization for inertial par-ticles moving in a periodic velocity field, and subject tomolecular diffusion. We show that, under appropriate as-sumptions on the velocity field, the large scale, long time

DS05 Abstracts 121

behavior of the inertial particles is governed by an effectivediffusion equation for the position variable alone. An ex-pression for the diffusivity tensor is found and various of itsproperties studied. Incompressible and potential fields arestudied, as well as fields which are neither, and theoreticalfindings are supported by numerical simulations.

Andrew StuartUniversity of [email protected]

Grigorios PavliotisDepartment of MathematicsImperial [email protected]

Martin HairerWarwick University, United [email protected]

MS25

Homological Characterization of Spiral DefectChaos

Relating the global structure of patterns exhibited to un-derlying dynamics is an important aspect of the study ofcomplex systems. We use homological characterizations tobuild insights into the dynamics of spiral defect chaos, aweakly turbulent state of Rayleigh-Benard convection. Wenote observations implying asymmetries between hot andcold flows, novel measures of boundary influence and in-dicators of system control parameters. We also find theevolution of the global structure of the flow to be primar-ily stochastic unlike the locally chaotic signatures reportedpreviously.

Kapil KrishanGeorgia Institute of TechnologySchool of [email protected]

Michael SchatzCenter for Nonlinear Science & School of PhysicsGeorgia Institute of [email protected]

MS25

Pattern Classification in Controlled Drug DeliverySystems

In an effort to maintain optimal levels of drug in the bodyover time, drug delivery systems have been developed thatfacilitate a sustained or periodic release of drug. The per-formance of these systems is significantly influenced by themicrostructure. Thus, a diffuse interface model is usedto predict microstructure formation and release behavior.Classification of the complex patterns that comprise themicrostructure in these systems enables us to establish rig-orous, quantitative structure-property relationships.

David M. SaylorFood and Drug [email protected]

James WarrenNational Institute of Standards and TechnologyMaterials Science and Engineering [email protected]

MS25

The Topology of Interfaces in Systems UnderCoarsening

We examine the evolution of the morphology of a two-phasemixture during coarsening under Cahn-Hilliard (CH) andAllen-Cahn (AC) dynamics. The morphology is quanti-fied using the interfacial shape distribution, which givesthe probability of finding a small patch of interface with acertain mean and Gaussian curvature. The scaling proper-ties of interfacial shape distributions and the differences inthese distributions for systems coarsening under CH andAC dynamics will be discussed.

YongWoo KwonDept. of Materials Science and EngineeringNorthwestern [email protected]

Katsuyo ThorntonDept. of Material Science and EngineeringUniversity of [email protected]

Peter VoorheesNorthwestern UniversityDept. of Material Science and [email protected]

MS25

Residual Stress Networks in Polycrystalline Mate-rials: Their Origin and Character

Crystallographic texture in conjunction with anisotropiccrystalline properties profoundly influences ensemble phys-ical properties and macroscopic behavior of polycrystallinematerials. Recently, polycrystalline materials with crys-talline thermal expansion anisotropy were observed to de-velop residual-stress networks with a length-scale that sur-prisingly could encompass many grains. Microstructure-based finite-element simulations were used to elucidate theorigin of these stress networks in the orientation and mis-orientation distribution functions. Residual stress isosur-faces for these networks were characterized by topologicaland fractal metrics.

Thomas WeissUniversitaet [email protected]

Peter S. [email protected]

Edwin R. Fuller, Jr.NISTCeramics [email protected]

David M. SaylorFood and Drug [email protected]


122 DS05 Abstracts

MS26

Nontwist Systems with Many Degrees of Freedom:A Mean Field Approach

Nontwist systems have received considerable attention overthe last years. However, little is known about nontwistHamiltonians with many degrees of freedom. To addressthis problem we consider mean-field coupled, nontwistsymplectic maps. Mean field coupled Hamiltonians haveproved to be a useful laboratory to study large degreesof freedom systems. Here we focus on the role of self-consistent dynamics in the formation of coherent struc-tures, separatrix reconnection, and the robustness of theshearless curve.

Diego Del-Castillo-NegreteOak Ridge National [email protected]

MS26

Vanishing Twist Near Focus-Focus Points

We show that near a focus-focus value in a Liouville in-tegrable Hamiltonian system with two degrees of freedomlines of locally constant rotation number in the image ofthe energy-momentum map are spirals determined by theeigenvalue of the equilibrium. From this representation ofthe rotation number we derive that the twist condition forthe isoenergetic KAM condition vanishes near the focus-focus point. This implies, e.g., that near the HamiltonianHopf-bifurcation the twist always vanishes.

Holger R. DullinLoughborough UniversityMathematical [email protected]

San Vu NgocUniversite GrenobleInstitut [email protected]

MS26

Normal Forms for Fold and Cusp Singularities forSymplectic Maps

An integrable symplectic map is characterized by its fre-quency mapping J → Ω(J). If Ω(J) is not one-to-one,the singularities give rise to “twistless bifurcations” uponperturbation. We study the normal forms in 4D for foldand cusp singularities in the neighborhood of a rationalrotation vector. A critical parameter is the slope of thecritical set in J . When the perturbation is small enoughthe map can be converted into a Hamiltonian flow that hasan approximate second invariant.

James D. MeissUniversity of ColoradoDept of Applied [email protected]

Holger R. DullinLoughborough UniversityMathematical [email protected]

A.V. IvanovSt. Petersburg UniversityRussia

[email protected]

MS26

Nontwist Hamiltonian Systems: An Introduction

In this talk, I will give an introduction to the study ofnontwist Hamiltonian systems with particular emphasis onapplications in physics. A simple example of such a sys-tem is the so-called standard nontwist map which has beenused in several physical models to study the transition toglobal chaos. I will discuss some recent work on invariantmanifold reconnection in this map.

Alexander WurmDepartment of Physics, Fusion StudiesThe University of Texas at [email protected]

MS27

A Coupled-Oscillator Model with a ConservationLaw for the Rhythmic Amoeboid Movements

Experiments on the fusion and partial separation of plas-modia of the true slime mold Physarum polycephalum aredescribed, concentrating on the spatio-temporal phase pat-terns of rhythmic amoeboid movement. On the basis ofthese experimental results we introduce a new model ofcoupled oscillators with one conserved quantity. Simula-tions using the model equations reproduce the experimen-tal results well.

Ryo KobayashiDepartment of Mathematical and Life SciencesHiroshima [email protected]

MS27

Introduction: Past Key Findings and Present Di-rections

I mention two key findings on collective motion of cellu-lar rhythms: 1) modes of phase difference in coupled someoscillators (Takamatsu), 2) control of taxis by using phasewave propagation. Phase equations are basically success-ful to analyze these phenomena. A recent challenge isdone to include significant roles of visco-elasticity of cell(Kobayashi). Behavioral intelligence of cell is also testedby posing some geometrical puzzles (Takagi) and analyzedby the mathematical model.

Toshiyuki NakagakiResearch Institute for Electronic ScienceHokkaido [email protected]


Atsushi TeroHiroshima [email protected]

DS05 Abstracts 123

MS27

Decision Making By True Slime Mold

The plasmodium of Physarum polycephalum is a naked ag-gregate of protoplasm, and this living mass migrates to seeka proper environment under various stimuli. The mecha-nism of decision making to a conflicting stimulation in theplasmodium was studied by analyzing the behavioral re-sponses to a mixture of attractants and repellents. Theavoidance to repellents shifted to the higher concentrationsas the attractant concentration became the higher. Onthe other hand, the membrane potential deflection variedsimilarly both in the presence and absence of attractants.Thus, the interference took place not at the receptor mem-brane, but at the signal integration. The mode of inter-ference was explained quantitatively by assuming that twokinds of signaling substances, corresponding to attractionand avoidance, bind competitively to a protein.

Tetsuo UedaHokkaido [email protected]

Seiji TakagiResearch institute for Electronic ScienceHokkaido [email protected]

MS27

Well-Controlled Biological Oscillator Systems forTrue Slime Mold Constructed with Microfabrica-tion Techniques

Oscillating cells of plasmodium of true slime mold can beconsidered as a coupled oscillator system. For syntheticand systematic analysis, we constructed coupled oscillatorsystems with living cells of plasmodium by cell patteringmethod using microfabrication technique, where configura-tion of oscillator parts and couplings can be systematicallycontrolled. Rich oscillation patterns and switching behav-ior among the patterns observed in systems consist of morethan three oscillators will be reported.

Atsuko TakamatsuDepartment of Electrical Engineering and BioscienceWaseda Universityatsuko [email protected]

MS28

Stochastic Stimulation of the Mammalian Circa-dian Clock

Based on an experimental estimate of the number ofmolecules of key proteins within the mammalian circadianclock, we simulate our model of the mammalian circadianclock with stochastic molecular interactions. Interactionswith promoters on the time scale of seconds are requiredfor accurate 24-hour timekeeping. The stochasticity of ourmodel scales in an expected way. We also find noise in-duced oscillations. This work was conducted with CharlesPeskin.

Daniel B. ForgerNew York [email protected]

MS28

A Stochastic Reaction-Diffusion-Active Transport

Method for Studying the Control of Gene Expres-sion in Eukaryotic Cells

A method is developed for incorporating diffusion andactive transport of chemicals in complex geometries intostochastic chemical kinetics simulations. Systems are mod-eled using a master equation, with jump rates for diffusivemotion and active transport between mesh cells calculatedfrom the discretization weights of an embedded boundarymethod. Since jumps between cells are treated as first or-der reactions, individual realizations of the stochastic pro-cess can be created by the Gillespie Method. The methodis used to study a 3D model of transcription, nuclear ex-port, translation, nuclear import, and gene regulation in aeukaryotic cell.

Charles S. PeskinCourant Institute of Mathematical SciencesNew York [email protected]

Samuel A. IsaacsonCourant Institute of Mathematical [email protected]

MS28

Reconstructing Subpopulation ConnectivityWithin Neuronal Networks

I present a mathematical framework for analyzing connec-tions among a subset of measured neurons embedded in alarger neural network. Through analyzing a simple prob-abilistic model of neural response, I demonstrate how toaccount for the presence of unmeasured neurons. One candetermine connectivity patterns in terms of certain neu-ral subpopulations, which are groups of neurons with asimilar response to a stimulus. Although the results arepresented in terms of neuronal networks, the mathemati-cal framework is applicable to other networks, such as generegulatory networks.

Duane NykampUniversity of MinnesotaSchool of [email protected]

MS28

The Influence of Chromosome Flexibility onAnaphase A Chromosome Movement as Driven byan Imperfect Brownian Ratchet Molecular Motor

The segregation of sister chromatids to daughter cells wasone of the earliest observations in cell biology. However,the mechanism by which they move has proven elusive,as no conventional motor protein seems wholly responsi-ble for the motion. We propose a Brownian ratchet modelas the driving mechanism behind chromosome segregation.We find that one determinant of the mean velocity in thissystem is the flexibility of the chromosome itself. More-over, as the ratchet becomes more ”perfect”, this effect isenhanced. Also, we find that a system with multiple Brow-nian ratchets allows for some manner of load independence,perhaps pointing to a resolution of the long standing ques-tion of why long and short chromosomes move at similarvelocities.

Arjun RajNew York [email protected]

124 DS05 Abstracts

MS29

Competition Between Mixing and Segregation inGranular Tumblers

Flowing granular materials tend to segregate, ordering con-stituents by size, density or even surface properties. Likefluids, chaotic advection in granular flows can enhance mix-ing. We investigate the competition between segregationand mixing of glass beads in unique tumbler geometriesthat produce a variety of patterns. A simple continuummodel the underlying flow gives insight into the structureof these segregation patterns.

James GilchristLehigh [email protected]

MS29

Reactions in Open 3D Flows

We study the dynamics of reactive particles advected by3-dimensional open incompressible flows, both analyticallyand numerically, using a 3D generalization of the bakermap. We find that 3D reactive flows have fundamentallynovel dynamical features, not found in 2D systems, due tothe richer structure of the unstable manifold in 3 dimen-sions. In particular, we show that the reaction output in3D displays features not found in 2-dimensional systems.

Alessandro MouraInstitute of PhysicsUniversidade de Sao [email protected]


MS29

On Chaotic Advection In 3-D Open Flows

Chaotic advection is examined in two 3-D open flows whichdiffer by their cross-section, the eccentric helical annularmixer (EHAM) and the 3-D flow between two confocalgliding ellipses. In both cases the outer boundary turns atconstant speed while the inner boundary turns at a timeperiodic angular velocity. Conditions leading to the bettermixing of an injected dye can be determined by analysis.Unexpectedly, results show that better mixing is obtainedin EHAM.

Antonio RodrigoUniversidade Nova de [email protected]

Esteban SaatdjianENSIC - INPL [email protected]

Paulo MotaUniversidade Nova de [email protected]

MS29

Mixing in Stokes Flows: From a Dynamical System

to the Diffusion Equation

We discuss the Lagrangian chaos in a particular Stokes flowthat is a superposition of a regular flow and a weak per-turbation. We demonstrate that the adiabatic invariant ofthe flow undergoes quasi-random jumps when a streamlinecrosses the resonance of the unpertubed flow. We showthat for multiple resonance crossings the accumulation ofthe jumps leads to the chaotic advection and mixing, therate of which is governed by a standard diffusion equation.

Anatoly NeishtadtSpace Research Institute,[email protected]

Igor Mezic, Dmitri L. VainchteinUniversity of California, Santa [email protected], [email protected]

MS30

Propagation in Neural Fields: an Introduction

In this introduction I will review the way in which synapti-cally coupled neural networks may generate and maintaintravelling waves of activity, whether they be in so-calledspiking or rate based models. For simplicity I will focuson waves in integrate-and-fire spiking networks and theiranalogues in firing rate models. This will set the scenefor the content of the two subsequent sessions, which willcover descriptions of real neural waves from the entorhinalcortex, the effect of plastic synaptic connectivity and ax-onal delays on wave stability, wave generation via externalstimuli, and instabilities in heterogeneous networks.

Stephen CoombesUniversity of [email protected]

MS30

Activity Propagation in Neural Fields for GeneralSynaptic Connections

The spatiotemporal dynamics of neural populations has at-tracted much attention in recent years. By virtue of thespatial extention of the population, transmission delay andnonlocal constant feedback delay are present and may af-fect the stability of the population. The presented workexamines the propagation of neural activity for generalhomogeneous connectivities involving transmission delay.In particular, conditions for plane waves are derived andthe effect of corticothalamic feedback on traveling fronts isstudied.

Axel HuttApplied Stochastic Processes, Humboldt University [email protected]

MS30

Rhythms and Propagating Activity in the Entorhi-nal Cortex

The entorhinal cortex plays an instrumental role in mem-ory formation while acting as an interface between the hip-pocampus and the neocortex. Theta and gamma frequencyrhythmic oscillatory activity of pyramidal cells, interneu-rons, and stellate cells in layers II, III, and V are thoughtto play an important role. Using biophysically based mod-

DS05 Abstracts 125

els of these cell types and dynamical systems techniques,we analyze the basic mechanisms for the formation, sup-pression, and propagation of these rhythms.



Jozsi Z. JalicsCenter for BioDynamicsBoston [email protected]

MS30

Spiral Waves in Neural Field Equations

Neural field equations are non-local PDEs thought to de-scribe large-scale dynamics of the cortex in various situa-tions. We discuss the existence of rotating spiral waves insuch equations on a two-dimensional domain, and followthem via numerical continuation as various parameters arechanged. This gives insight into which physiological pa-rameters should be manipulated to either destroy or en-hance the stability of these waves.

Carlo R. LaingMassey [email protected]

MS31

Dynamics in Aerospace Applications: from Controlto Design of Dynamics

We will show how certain aspects of design of unsteady flowdevices for military aerospace applications contribute toorigin of detrimental oscillations. We will also ways of mit-igation of oscillations by active and passive control as wellas by modification of design, including symmetry breaking.The analysis of the oscillations will be done using nonlin-ear distributed models involving multiple oscillatory modescoupled through nonlinear terms and distributed transportdelay, that are driven by broad-band disturbances.

Andrzej BanaszukUnited Technologies Research [email protected]

MS31

Optimal Control of Formation Flying Satellites

Future space missions like Terrestrial Planet Finder(NASA) and Darwin (ESA) will make use of a networkof formation flying spacecraft. In these missions, the re-quirements on the accuracy of the relative positioning ofthe craft are extremely high. In addition, reconfigurationsof the formation have to be performed at regular intervalswith minimal energetical effort. In this talk, we show howa recently developed variational method for the numericalcomputation of optimal open-loop controls for mechanicalcontrol systems can be applied to this problem. We out-line the method, its benefits and present example compu-

tations.

Oliver JungeInstitute for [email protected]

Jerrold E. MarsdenControl and Dynamical SystemsCalifornia Institute of [email protected]

Sina Ober-BlobaumInstitute for MathematicsUniversity of [email protected]

MS31

Discrete Mechanics, Variational Integrators, andOptimal Control

The theory of discrete mechanics and associated variationalintegrators is applied to optimal control of mechanical sys-tems in which one attempts to optimize a cost functionsubject to equations of motion in mechanics. The novelfeature of our approach is that we use discrete mechan-ics to represent the equations of motion as algebraic con-straints. We show that this is especially useful in problemsin which one has long time problems (as in low thrust) orin which one needs to take large time steps. The method isillustrated with satellite control, control of a group of hov-ercraft and coordinated control of a group of underwatervehicles.

Sina Ober-BloebaumUniversity of [email protected]



MS31

Multicomponent Dynamical Systems and GraphTheory

We discuss asymptotic behavior and uncertainty propaga-tion in nonlinear dynamical systems on graphs. We utilizedigraph decomposition theory of the system to prove sev-eral results on asymptotic dynamics of large systems thatdepend only on their structural properties. We discuss dis-turbance propagation in such systems and show that itdepends on the node distance in the associated graph. Weconnect the theory to issues of recent interest in industrialcontext.

Igor MezicUniversity of California, Santa [email protected]

MS32

A Unified Lyapunov Function Approach for Nonau-

126 DS05 Abstracts

tonomous Attractors

In this talk we present a framework for Lyapunov functionsfor nonautonomous attractors. We show that one and thesame Lyapunov function construction yields a characteri-zation of (i) pullback attractors (ii) forward attractors and(iii) uniform attractors. Our construction is based on ideasgoing back to Yoshizawa combined with a careful analysisof the attraction rates expressed via comparison functionswhich generalize the concept of KL functions widely usedin nonlinear control theory.

Stefan SiegmundUniversity of [email protected]

Fabian WirthUniversity of [email protected]

Lars GrueneMathematical Institute, University of [email protected]

Peter KloedenDepartment of MathematicsJ.W. Goethe University [email protected]

MS32

Global Behaviour in Adaptive Control Systems

Adaptive controllers are used in systems where one or moreparameters are unknown. Such controllers are designedto stabilize the system using estimates for the unknownparameters that are adapted automatically as part of thestabilization. One drawback in adaptive control design isthe possibility that the closed-loop limit system is not sta-ble. The worst situation is the existence of a destabilizedlimit system attracting a large open subset of initial con-ditions. These situations lie behind bad behavior of theclosed-loop adaptive control system. In this talk we iden-tify and characterize the occurrence of such bad behaviorin the adaptive stabilization of first- and second-order sys-tems with one unknown parameter. In this context we dis-cuss a number of bifurcation-like phenomena and developcorresponding normal forms.

Hinke M. OsingaUniversity of BristolDepartment of Engineering [email protected]

Reza Rokni LamookiUniversity of MazandaranBabol, [email protected]

Stuart TownleyUniversity of ExeterExeter, [email protected]

MS32

Dimension for Attractors for Non-Autonomous Dy-

namical Systems

The Hausdorff dimension is a parameter which describesthe behavior of dynamical systems. In particular, there areseveral method to estimate that dimension for the attractorof such a system. In the case of a non-autonomous systemthe so called pullback attractor gives a description of thedynamics. We will discuss several methods to estimate theHausdorff dimension of those attractors.

Bjorn SchmalfussUniversity of [email protected]

MS32

Vortex Merging

The merging of two vortices is described using tools fromnonautonomous dynamical systems theory. It is crucialto capture the hyperbolic transient behavior by finite-timeinvariant manifolds.

Stefan SiegmundUniversity of [email protected]

MS33

Distributed Coordination of Mobile Agents: FromBird Flocking to Synchronization of Coupled Os-cillators

In this talk we provide a unified view of several distributedcoordination and flocking algorithms which have appearedin various disciplines such as statistical physics, biology,computer graphics over the past 2 decades. These algo-rithms have been proposed as a mechanism for demon-strating emergence of collective behavior (such as socialaggregation, schooling, flocking and synchronization) us-ing purely local interactions. We will show that thesecoordination problems, such as Vicsek’s model of coor-dination of self propelled particles in statistical physics,The Reynolds’ model of flocking in computer graphics, andthe synchronization of coupled nonlinear oscillators (a wellstudied problem in dynamical systems and physics), can beall studied and analyzed rigorously under a unfied frame-work of graph theory and dynamics. Utilizing these results,we provide a biologically plausible, vision-based coordina-tion scheme for flocking and velocity alignment, which doesnot require velocity measurements and/or nearest neigh-bor communication, but instead relies on nearest neighborsensing. We will show that by sensing the optical flow andtime-to-collision between each agent and its neighbors wecan achieve coordination, even if the topology of the prox-imity graph changes with time. Finally, we use the sameframework to analyze a recently proposed scheme for geo-graphic routing in wireless adhoc networks which does notrely on location information.

Ali JadbabaieDepartment of Electrical and Systems EngineeringUniversity of [email protected]

MS33

Formations from Gyroscopic Interactions

We consider gyroscopic interactions among particles mov-ing in three-dimensional space, and demonstrate their ef-

DS05 Abstracts 127

fectiveness in producing desired stable spatial patterns ofmotion. Applications include formation control for teamsof agile vehicles such as UAVs. We discuss how best toframe the curves representing the particle trajectories. Anatural Lie group formulation emerges, with G = SE(3)as a symmetry group. For two-particle formations withspecific G-invariant interaction laws, we are able to proveglobal convergence to specific formations (as well as non-collision). We also describe how a single particle mightinteract with a fixed structure in space by exploiting gyro-scopic feedback to achieve obstacle avoidance or boundary-following behavior.

Eric W. JusthUniversity of [email protected]

Fumin ZhangPrinceton [email protected]

P.S. KrishnaprasadUniversity of Maryland, College ParkInstitute for Systems [email protected]

MS33

Self-Organizing Robotic Systems

The fundamental problem of designing self-organizing sys-tems is to understand how local interactions between com-ponents give rise to global properties. This problem mustbe solved if we are to engineer predicatable and reliablelarge-scale systems consisting of vast numbers of parts (e.g.micro-robots, cells or molecules). In this talk I will de-scribe a formal approach to modeling and designing self-organizing systems based on graph rewriting. The ap-proach allows us to describe massively parallel algorithmsfor self-assembly, self-replication, distributed locomotionand other decentralized processes and to rigorously provethat they work. I will illustrate the approach by show-ing how it can be used with our self-assembling roboticsplatform and in MEMs self-assembly.

Eric KlavinsDepartment of Electrical EngineeringUniversity of [email protected]

MS33

Self-Assembly and Collective Control of Agent-Based Swarms

Swarms of sensors and mobile agents can be used for mas-sive distributed sensing, monitoring, modeling, and map-ping of an environment. In this talk, we discuss a theoryfor modeling and control of collective motion of swarms ofparticles with local interactions. We present distributedswarming algorithms that allow a leaderless swarm of par-ticles to migrate from point A to B without inter-agent oragent-to-obstacle collisions. We view a ”swarm” as a com-plex network of dynamical systems. A class of semiregu-lar lattice structures called ”alpha-lattices” are introducedto represent the dynamic topology of inter-agent interac-tions. It turns out that self-assembly and landscape prop-erties of alpha-lattices are instrumental in analysis of ourswarming algorithms. Formal results are established onself-alignment and spatial order of particle-based swarms.Moreover, a multi-species particle system is used to design

and analyze the collective behavior of swarms in presenceof obstacles. We provide simulation results that success-fully demonstrate migration, split/rejoin, squeezing, andrendezvous maneuvers for swarms of hundreds of agents.

Reza Olfati-SaberUniversity of California, Los Angeles &California Institute of [email protected]

MS34

DNA in Flows with Controlled Mixing

We model the recombination of very-short-strand DNAwith the assumption that two complementary strands willcombine only if they are close together in position as wellas alignment. We describe this as a reaction-advection-diffusion system in position-orientation space with avenuesfor control in the form of velocity fields and external po-tentials. We analyze the dynamics in a flow caused by anactive (shear superposition) micro-mixer and compare withexperiments.

Igor Mezic, Thomas JohnUniversity of California, Santa [email protected],[email protected]

MS34

Geometry and Addressability Actuation in Cat-alytic Pattern Formation

We investigate the effect of two-dimensional composite ge-ometry on the dynamic behavior of pulses through com-bined experimentation and modeling. With a ”Y”-shapestructure, we show how to direct the propagation of pulsesby varying the details of the geometry. Then we extend ourwork to numerical bifurcation analysis on pulse propaga-tion in a two-dimensional ring structure using matrix-freetechniques. The combination of confining geometry andlaser addressability on CO oxidation pulses is also explored.

Harm-Hinrich RotermundFritz-Haber-Institute of the Max Planck SocietyBerlin, [email protected]

Christian PuncktFritz-Haber-Institute of the Max Plack SocietyBerlin, [email protected]

Liang QiaoDepartment of Chemical EngineeringPrinceton [email protected]

Yannis [email protected]

MS34

Addressable Excitable Media for Modeling Collec-tive Behavior

We discuss two topics of collective behavior in the con-text of excitable media models, swarming behavior andspatiotemporal networks. Studies of controlling reaction-

128 DS05 Abstracts

diffusion waves with realistic excitability potentials are de-scribed. We also describe a study of dynamical networksin the photosensitive Belousov-Zhabotinsky reaction. Wemodel local nearest-neighbor interactions by the spread ofreaction-diffusion waves, while nonlocal excitations are de-scribed by nondiffusive jumps along shortcuts defined inthe medium.

Kenneth ShowalterWest Virginia UniversityDepartment of [email protected]

MS34

Controlling Faraday Wave Interactions via Multi-frequency Forcing

Faraday waves form on the surface of a fluid when it issubjected to a sufficiently strong vibration. Experimentsperformed over the last decade have shown that the spatio-temporal form of the standing wave pattern depends on thefrequency content of the periodic forcing function. We usea framework of equivariant bifurcation theory to show howto design parametric forcing functions in order to enhanceor inhibit weakly nonlinear three-wave interactions in theFaraday system.

Mary C. SilberNorthwestern UniversityDept. of Engineering Sciences and Applied [email protected]

Chad TopazUniversity of California, Los [email protected]

Jeff PorterUniversidad Complutense [email protected]

MS35

Self-Similar Conformations in Glassy WrinkledMembranes: From Casacade-Like Wrinkling to aSub-Critical Compaction

Partially polymerized membranes display a striking me-chanical transition at low temperature known as the wrin-kling transition. Fluorescence and scanning electron micro-scope as well as profile measurements using an atomic forcemicroscope (AFM) revealed the existence of three degreesof wrinkling depending on the degree of the membranepolymerization. At low polymerization the membrane un-dergoes a cascade of wrinkling to form a folded phase witha roughness exponent η equal to 3, at intermediate poly-merization, the membrane is in an intermediate-wrinkledphase (similar to the crumpling of an elastic sheet) withη ∼ 2.5, while at high polymerization the membrane un-dergoes an abrupt and sub-critical “compaction” to thewrinkled-rough phase and η ∼ 2. These phases, into whichthe membranes lock, are glassy roughened conformationswith memory depending on the degree of polymerization.

Sahraoui ChaiebDept. of Theor. & Appl. Mech.Univ. of Ill., [email protected]

MS35

Self-Similarity in Burgers Turbulence

I will describe simple and optimal conditions for universal-ity and non-universality in Burgers turbulence (the statis-tics of the Cole-Hopf solution to Burgers equation with ran-dom initial data). The results depend on a striking con-nection between Burgers turbulence and Smoluchowski’scoagulation equation discovered by Bertoin and our classi-fication of dynamic scaling in Smoluchowski’s coagulationequation.

Robert L. PegoCarnegie Mellon UniversityDepartment of Mathematical [email protected]

Govind MenonApplied Mathematics, Brown [email protected]

MS35

Ultimate Dynamics on the Scaling Attractor forSmoluchowski’s Coagulation Equations

We describe a basic framework for studying dynamic scal-ing that has roots in dynamical systems and probabilitytheory, and consider coagulation equations with rate ker-nels K = 2, x + y, xy. We previously classified all self-similar solutions (fixed points under scaling) and their do-mains of attraction. Now we show the dynamics on the setof all limit points modulo scaling (the scaling attractor)is linearized by Bertoin’s Levy-Khintchine representationfor eternal solutions, and this dynamics is chaotic. Thescaling attractor contains a dense set of scaling-periodicsolutions, and solutions with dense trajectories, analogousto Doeblin’s universal laws in probability theory.

Robert L. PegoCarnegie Mellon UniversityDepartment of Mathematical [email protected]

Govind MenonApplied Mathematics, Brown [email protected]

MS35

Coarsening in Thin-Film Equations: Upper Boundon Coarsening Rate

Thin, nearly uniform layers of some fluids can destabilizeunder the effects of intermolecular forces. After the ini-tial phase, the fluid breaks into droplets connected by anultra-thin layer of fluid. This structure coarsens on slow-time scale. The characteristic distance between dropletsand their size grow, while their number is decreasing. Thisphysical process can be modeled in the lubrication approx-imation by a so called thin-film equation for the height ofthe fluid. I will discuss coarsening in thin-film equationswith mobility equal to the height of the fluid. These equa-tions are gradient flows in the Wasserstein metric. Usingthe gradient flow structure within the Kohn-Otto frame-work we obtain rigorous upper bound on the coarseningrate. The upper bound we obtain coincides with the coars-ening rate that Glasner and Witelski conjectured for the1-dimensional problem.

Dejan Slepcev

DS05 Abstracts 129

UCLAMathematics [email protected]

Felix Otto, Tobias RumpUniversity of [email protected], [email protected]

MS35

Coarsening Dynamics of the Convective Cahn-Hilliard Equation

Stephen J. WatsonEngineering Sciences and Applied MathematicsNorthwestern [email protected]

MS35

Viscous Entrainment: Singular and Nearly Singu-lar Spouts

A small air bubble rising in syrup remains spherical. Alarger air bubbles deforms, developing an increasingly ta-pered trailing end. For an even larger air bubble, the ris-ing movement so severely deforms the bubble that a thintendril of air is deposited behind the rising bubble. We an-alyze such entrainment dynamics in a simple model prob-lem where a long-wavelength model describes the essentialdynamics. We show that both continuous and weakly dis-continuous entrainment transitions are possible when theinterface shape on the largest length-scales is constrainedso that the base of the entrained tendril approaches a coni-cal shape. Finally, we show that two kinds of critical tran-sition exist in the full problem because the scale-invariantdynamics supports a saddle-node bifurcation. After thebifurcation, scale-invariant solutions which can link ontophysical large-scale conditions do not exist.

Wendy W. ZhangPhysics Department & James Franck InstituteUniversity of [email protected]

MS36

A Heuristic Explanation of a Trapping MechanismThat Results in Large Interspike Intervals in theHH Equations with Excitatory Coupling

Excitatory self coupling in a reduced Hodgkin Huxley sys-tem is explored. The excitatory coupling causes an unex-pected effect on a system where the neurons rapidly ’syn-cronize’ and maintain very long ISIs (inter spike intervals).The trapping mechanism is a vortex induced by the rela-tive movement of the slow synaptic variable and the slowinactivation variable, h. Unlike many canards, this occu-rance is not extremely sensitive to choice of parameters andpersists over an interval of parameter sets.

Jonathan DroverUniversity of PittsburghPittsburgh, PA, [email protected]

MS36

Canards for a Reduction of the Hodgkin-Huxley

Equations

It is shown that canards, which are periodic orbits forwhich the trajectory follows both the attracting and re-pelling part of a slow manifold, can exist for a two-dimensional reduction of the Hodgkin-Huxley equations,which model the generation of action potentials for a squidgiant axon. By smoothly connecting stable and unstablemanifolds in an asymptotic limit, the parameter value atwhich canards exist for this system are predicted with greataccuracy.


MS36

Resonance in Stellate Cells of the Medial Entorhi-nal Cortex: A Canard Mechanism?

Stellate cells in layer II medial entorhinal cortex exhibitsubthreshold resonance at theta frequencies (8-12 Hz).We study this phenomenon using a biophysical model ofHodgkin-Huxley type. In the subthreshold regime ourmodel reduces to a three-dimensional system with one fastand two slow variables. We show that the resonance effectis is the result of an underlying canard structure that alsogoverns intrinsic subthreshold oscillations. We generalizeour studies to noisy and spiking systems.


Andreas V.M. HerzHumboldt Universitaet zu [email protected]

Tim OppermannHumboldt Universitaet zu BerlinBerlin, [email protected]

Horacio G. RotsteinCenter for BiodynamicsBoston [email protected]

MS36

Subthreshold Oscillations and Spiking in a MedialEntorhinal Cortex Stellate Cell

Medial entorhinal cortex stellate cells (SC) develop low-amplitude rhythmic subthreshold oscillations at theta fre-quencies (8-12 Hz) when depolarized to a value below spik-ing threshold. As the membrane potential approaches spik-ing threshold, SCs fire action potentials at the peak of theoscillations but not necessarily at every cycle. In this workwe show that the observed mixed-mode oscillations are gen-erated via a canard mechanism in a SC model having a two-component hyperpolarization-activated and a persistent-sodium currents


130 DS05 Abstracts


MS36

Panel Discussion


Martin WechselbergerThe Ohio State [email protected]

MS36

Significant Slowing of Firing Rates in NeuronsThrough the Canard Phenomenon

Recent work on Hodgkin Huxley type neurons respectivelyneural networks showed a significant slowing of the firingrate under certain cirumstances. This slowing of the firingrate is accompanied by subthreshold oscillations near theaction potential threshold. I show that canards are respon-sible for that delay and line out how to identify this canardphenomenon in biophysical problems.

Martin WechselbergerOhio State UniversityMathematical Biosciences [email protected]

MS37

Channelling Chaos by Building Barriers

Chaos often represents a severe obstacle for the set-up ofmany-body experiments, e.g., in fusion plasmas or turbu-lent flows. We propose a strategy to control chaotic dif-fusion in conservative systems. The core of our approachis a small apt modification of the system which channelschaos by building barriers to diffusion. It leads to practicalprescriptions for an experimental apparatus to operate ina regular regime (drastic enhancement of confinement).

Cristel ChandreCentre de Physique Theorique - CNRSMarseille, [email protected]

MS37

Reaction Rates of Chemical Systems with Three orMore Degrees of Freedom

We merge the ideas of tube dynamics and reaction is-land theory into a comprehensive theory of chemical re-action rates that overcomes some of the problems plagu-ing the Transition State Theory. By computing normallyhyperbolic invariant manifolds and their stable and unsta-ble manifolds, and merging them with Monte Carlo meth-ods to compute the volumes of tube intersections withinPoincare sections, we provide the theoretical and computa-tional tools for computing accurate chemical reaction ratesin realistic systems.

Frederic Gabern

[email protected]

Shane [email protected]

Wang Sang KoonControl and Dynamical SystemsCalifornia Institute of [email protected]


MS37

Controlling Mixing in Three-Dimensional, Volume-Preserving Mappings

The motion of a passive scalar in time-periodic, incom-pressible fluid flows, such as those in certain Rayleigh-Bernard instabilities, can be modeled by volume-preservingmappings. Here we investigate a class of flows we call”blinking-rolls” and the optimization of mixing behaviorin these maps under changes in their parameters.

Keith JulienUniversity of [email protected]

James D. MeissApplied [email protected]

Paul MullowneyUniversity of [email protected]

MS37

Dynamics and Control of Macromolecules

We study a network of nearest-neighbor coupled oscillatorsstarting with a simple coarse-grained model of a moleculewith a backbone and side-chains. We show that this sys-tem is particularly good at reacting responsively to local-ized disturbances by amplifying them. We present analysisof an interesting transition phenomenon between global en-ergy minima of the system and relate this to recent resultson controllability of Hamiltonian systems. More generally,oscillator networks that exhibit such phenomena are char-acterized by nearest neighbor interactions of a node thatare, in most of the phase space, much stronger than thenonlinear oscillations of the local dynamics at the node.

Igor MezicUniversity of California, Santa [email protected]

MS37

Control of Resonances in Hamiltonian Systems

To achieve large changes in adiabatic invariants usingsmall control inputs, a conservative dynamical system mustposses an internal resonance. We propose a control methodto use capture into resonance to transport particles. When

DS05 Abstracts 131

the nominal dynamics brings particles close to a resonancesurface, a control pulse forces the capture of particles intoresonance. Captured particles is transported across the en-ergy levels and then released when the desired energy levelis achieved.

Dmitri L. Vainchtein, Igor MezicUniversity of California, Santa [email protected], [email protected]

MS38

Spatiotemporal Dynamics as a Function of Connec-tivity in Cardiac Cell Cultures

Monolayer cultures of cardiac cells form a biological ex-citable medium that display different spatiotemporal pat-terns, ranging from target patterns to multiple spiralwaves, depending on experimental conditions. Monolay-ers are optically mapped under different concentrationsof pharmacological agents that alter cell-cell connectivity.Spatiotemporal dynamics are then characterized by an en-tropy measurement, which facilitates comparison to a sim-ple theoretical model.

Gil BubSUNY Health Science Center at [email protected]

MS38

Parallel Implicit Methods for the Bidomain Equa-tions

In the bidomain model of cardiac electrophysiology, cur-rent conservation takes the form of an elliptic constrainton the potentials defined for the intracellular and extracel-lular compartments. To satisfy this constraint, a system ofequations must be solved at each timestep, making implicittime discretizations a natural approach. We describe bothlinearly and nonlinearly implicit schemes for the bidomainequations, discuss their comparative accuracy and efficientparallel implementation, and present fully resolved three-dimensional computed dynamics.

Charles S. PeskinCourant Institute of Mathematical SciencesNew York [email protected]

Boyce E. GriffithNew York [email protected]

MS38

Mathematical/computer Modeling of the Effects ofInhomogeneities on Cardiac Defibrillation

Cardiac fibrillation is the deterioration of the heart’s nor-mally well organized activity into one or more meanderingspiral waves which subsequently break up into many me-andering wavefronts. If it is not treated immediately itleads to death. The only way to stop fibrillation is bythe application of an electric shock (defibrillation). Thisstudy focuses on examining whether higher degrees of dis-organization of the fibrillation has an effect on the shockstrength required to defibrillate, and whether microscopicconductivity fluctuations favor shock success. We devel-oped a three dimensional computer bidomain model of a

block of cardiac tissue with straight fibers immersed in aconductive bath. Intracellular conductivities were variedstochastically around nominal values with variations of upto 50%. A single rotor reentry was initiated and, by ad-justing the spatial variation of action electrical parametersthe level of organization could be controlled. The singlerotor could be stabilized or spiral wave breakup could beprovoked leading to fibrillatory-like activity. For each levelof organization, multiple shock timings and strengths wereapplied to compute the probability of shock success as afunction of shock strength. Our results suggest that thelevel of the small-scale conductivity fluctuations is a veryimportant factor in defibrillation.

L. Joshua LeonUniversity of Calgary, [email protected]

Edward VigmondDepartment of Electrical and Computer EngineeringUniversity of [email protected]

Gernot PlankKarl Franzens University at Graz, Graz, [email protected]

MS38

The Dynamics of Multiple Spiral Waves As aStochastic Predator-Prey System

We have found that a system containing many unstablespiral waves can be modeled as a stochastic predator-preysystem. If cells in the excited state are considered predatorsand cells in the recovered state prey, circular orbits in phasespace consistent with predator-prey behavior are observed.The mean time to spiral wave extinction is much largercloser to the inferred fixed point, suggesting a predictiverole for the predator and prey quantities.

Sandeep MannavaDepartment of BiologyCornell [email protected]

Robert F. Gilmour Jr.Department of Biomedical SciencesCornell [email protected]

Niels F. OtaniDepartment of Biomedical ScienceCornell [email protected]

MS38

Subcellular Turing Instability Mediated by Voltageand Calcium Diffusion in Cardiac Cells

We investigate the spatiotemporal dynamics of subcellularcalcium alternans within cardiac cells. We show the exis-tence of a pattern forming instability that leads to spatiallydiscordant calcium alternans. This instability is mediatedby the diffusion of membrane voltage and intracellular cal-cium, and is found to rely on the bidirectional couplingbetween these species. We describe the conditions for thisinstability to occur, and present a mathematical descrip-

132 DS05 Abstracts

tion of the pattern formation process.

Yohannes ShiferawUCLA School of [email protected]

MS38

Field Stimulation of Heart Tissue: Surface Polar-ization Versus Intramural Virtual Electrodes

The mechanism by which electric fields terminate arrhyth-mias continues to puzzle investigators. Existing experi-mental methods provide information about epicardial man-ifestations of electrical cardioversion, yet little is knownabout field effects inside the myocardium. We combinedspecially designed optical mapping experiments and com-puter modeling to separate the intra-myocardial and sur-face field effects. We find only minor surface polarizationduring field stimulation. Intramural virtual electrodes pro-duced even by weak fields are sufficiently strong to initiateintra-myocardial excitation.

Christian Zemlin, Arkady PertsovSUNY Upstate Medical [email protected], [email protected]

MS39

Energy Spectra of the Ocean’s Internal Wave Field:Theory and Observation

Yuri V. LvovRensselaer Polytechnic [email protected]

MS39

Nonlinear Stability of Stratified Shallow Water Dy-namics

Paul A. MilewskiDept. of MathematicsUniv. of [email protected]

MS39

Effects of Rotation on Breaking Waves

Ruben [email protected]

MS39

Near Resonances and Symmetry Breaking in Ro-tating Turbulence

Leslie SmithMathematicsUniversity of [email protected]

MS39

Stability of Stratified Sheared Flows

We contrast two distinct scenarios for mixing of stratified

flows: shear instability and breaking waves. In the con-text of hydrostatically balanced flows, these correspond totransitions to elliptic behavior of an otherwise hyperbolicsystem, and nonlinear wave steepening. We ask whethershear instability can occur at all in unforced flows. Alongthe way to answering this basic question, surprising fea-tures of the hierarchy of models describing hydrostaticallybalanced flows, from single-layered to continuously strati-fied, are revealed.

Esteban G. TabakCourant InstituteNew York [email protected]

MS40

Travelling Solitons in the Discrete NonlinearSchroedinger Equation

Asymptotic methods are used to study moving discretesolitons on the line, with particular attention to the ex-istence of radiation tails. We also study discrete periodicwaves on finite intervals.

Oliver Oxtoby, Igor BarashenkovUniversity of Cape [email protected], [email protected]

MS40

Embedded 2pi-Kinks in the Generalised Sine-Gordon Lattice: A New Barrier

An explanation is offered for an observed lower bound onthe wave speed of travelling kinks in Frenkel Kontorovalattices. Kinks exist at discrete wavespeeds within a pa-rameter regime where there is resonance with linear waves(they are embedded solitons). However, they fail to existwhenever there is more than one branch in the dispersionrelation. Novel numerical methods are used to continuecurves of 2π- and 4π-kinks in propagation speed, latticediscreteness and a tunable amount of onsite anharmonicity.This is joint work with Andre Aigner and Vasillis Rothos.


MS40

Stationary and Travelling Kinks in the DiscretePhi-Four Model

We introduce several new discretisations of the cubic Klein-Gordon equation (the so-called phi-four theory) which ex-hibit continuous families of (stationary) kink solutions. Westudy the existence of travelling kinks in these models, aswell as in discretisations proposed previously. Our methodis based on the asymptotic analysis beyond all orders ofthe perturbation theory.

Dmitry PelinovskyMcMaster University, [email protected]

Oliver Oxtoby, Igor BarashenkovUniversity of Cape [email protected], [email protected]

DS05 Abstracts 133

MS40

Classical and Quantum Peierls-Nabarro Barriers

Usually, discrete solitons experience a potential barrier asthey travel through the lattice, which can induce fast ra-diative deceleration and trapping. We will review variousdiscrete systems where this barrier is totally absent, de-scribing their possible physical applications, then demon-strate that, even in the absence of a classical barrier, thephonon Casimir energy induces a quantum energy barrierof generically similar type. Thus a purely quantum kinktrapping mechanism arises in these systems.

James M. SpeightSchool of MathematicsUniversity of [email protected]

MS40

Stokes Constant for Discrete Maps

Alex TobvisDepartment of MathematicsUniversity of Central [email protected]

MS40

Moving Breathers in One-Dimensional FPU andTwo-Dimensional FPU-Klein-Gordon Lattices

We seek moving breather solutions in one-dimensionalFermi-Pasta-Ulam (FPU) chains with cubic-quartic poten-tial energy functions. We find a family of solutions whichcan be continuously varied from traditional breathers tobreathing-kink travelling waves. The breathing-kink solu-tions have a kink amplitude which can be arbitrarily small,(the quartic FPU lattice has a threshold below which noclassical kinks exist). Our approximations use multiplescales asymptotic techniques, which we then use to inves-tigate breathers in two-dimensional systems.

Jonathan Wattis, Imran ButtSchool of Mathematical SciencesUniversity of [email protected],[email protected]

MS41

Bifurcations in Neural Fields with Delays

A class of integro-differential equations modelling neuralactivity is considered which take into account the finitespeed of transmission along axons and cortico-thalamicfeedback delays. The stability of spatially homogeneousequilibria and the bifurcations leading to spatial patterns,oscillations, and traveling waves are investigated. A per-turbative method is presented which allows the determina-tion of the bifurcations for arbitrary interaction kernels.

Fatihcan M. AtayMax Planck Institute for Mathematics in the Sciences,[email protected]

MS41

Spectro-Temporal Defects Induced By Advection

in a Globally Coupled System

We evidence numerically and experimentally that advec-tion can induce spectrotemporal defects in a system witha localized solution. We start with the study of a free-electron laser, which is described by a one-dimensionalconvection-diffusion model with global saturation coupling.Then, we show that the main features of the instability arekept in simple Ginzburg-Landau equations with advection.The situations of local and global coupling are compared.

Christelle Bruni, David Garzella, Gian-Luca Orlandi,M.-E. CouprieCEA/[email protected],[email protected],[email protected], [email protected]

Christophe SzwajUniversite de Lille (France)Laboratoire [email protected]

Serge Bielawski

PhLAM/Universite Lille I,[email protected]

MS41

Bumps, Breathers, and Waves in a Neural Networkwith Threshold Accommodation

I will discuss a continuum model of neural tissue that in-cludes the effects of accommodation. The basic model is anintegral equation for synaptic activity that depends uponthe non-local network connectivity, synaptic response, andfiring rate of a single neuron. A phenomenological modelof accommodation is examined whereby the firing rate istaken to be a simple state-dependent threshold function.As in the case without accommodation classical Mexican-Hat connectivity is shown to allow for the existence of spa-tially localised states (bumps). Importantly an analysisof bump stability using recent Evans function techniquesshows that bumps may undergo instabilities leading to theemergence of both breathers and travelling waves (and in-deed travelling breathers). Numerical simulations showthat bifurcations in this model have the same generic prop-erties as those seen in many other dissipative systems thatsupport localised structures, and in particular those of cou-pled cubic complex Ginzburg-Landau equations and threecomponent reaction diffusion systems.

Markus Owen, Stephen CoombesUniversity of [email protected],[email protected]

MS41

Stability of Nonlocal Neural Fields Subject to Ad-ditive Stochastic Forces

The spatiotemporal dynamics of neural populations has at-tracted much attention in recent years. By virtue of thenetwork of axonal connections between neurons, there arenonlocal interactions yielding various spatial patterns. Thepresented work discusses the stability of the neural fieldsubjected to general external stimulus. Further the emer-gence of the Turing instability is discussed subjected to

134 DS05 Abstracts

stochastic forces. We derive the order parameter equationand find stability threshold shifts dependant on the noiselevel.

Axel HuttApplied Stochastic Processes, Humboldt University [email protected]

Lutz Schimansky-GeierInstitute for PhysicsHumboldt University at [email protected]

MS41

Q-Switching Instability in Passively Mode-LockedLasers

A new asymptotic analysis of the equations for a mode-locked laser exhibiting a Q-switching instability is pro-posed. The averaging of the fast time in the derivationof the model introduces a non-local term in the equationfor the population inversion. We show that this non-localterm leads to a Hopf bifurcation and we derive analyticalthresholds for it. Our analysis is valid for all class B lasersexhibiting slowly damped relaxation oscillations which in-clude solid state and semiconductor lasers. Our analyticalresults are shown to compare favorably with numerical sim-ulations.


MS41

Localized Vegetation Structures in Arid Land-scapes

Localized vegetation structures are large-scale vegetationpatterns observed worldwide in arid and semiarid territo-ries. Single localized vegetation structure consists of eitheran isolated patch surrounded by a bare state or a bare spotin sparsely vegetated area. Their spatial distribution canbe either ordered or random. A model and a non-linearanalysis are presented to account for their formation andstudy their interactions.

Mustapha Tlidiuniversite Libre de BruxellesOptique nonlieaire [email protected]

MS42

Multimode Dynamics of a Transmission Line Os-cillator

A transmission line oscillator consists of a length of trans-mission line terminated with a negative resistor and adiode. The complexity of the dynamics of this oscillatordepends on the number of active modes in the line. Witha fixed bandwidth and a very short transmission line, onlya single mode is active. A longer line allows multiple modesto oscillate and interact. We present an experimental in-vestigation of a line with two active modes.

Ned J. Corron, Shawn PethelU.S. Army [email protected], [email protected]

Jonathan N. BlakelyUS Army [email protected]

MS42

Beam Steering by Lag Synchronization in Wide-band Chaotic Arrays

Chaotic oscillators are an intriguing source of waveformsfor ultra wideband-radar applications. The broadband andnonrepeating nature of chaos provides an ideal combinationof high range resolution and zero range ambiguity. Arraysynchronization via local coupling is an efficient alternativeto the use of a master oscillator in a phased array; however,a difficulty in any wideband array is achieving a practical,cost-effective mechanism for beam steering. Here, we ex-plore the use of lag and anticipating synchronization in achaotic array to achieve steering. The natural frequency ofindividual oscillators is adjusted to control a uniform lagacross the array. The direction and extent of the steer-ing is directly controlled by the orientation and magnitudeof the induced lag or anticipation. Using chaotic circuitsoperating at 20 MHz, we show that lag synchronizationcan be practically controlled in arrays of rf devices. Openquestions include determining stability and dynamical lim-itations for steering the larger arrays that practical appli-cation will require.

Ned J. Corron, Shawn PethelU.S. Army [email protected], [email protected]

Jonathan N. BlakelyUS Army [email protected]

MS42

Hopf Bifurcations in Time-Delay Systems withBand-Limited Feedback

We investigate the steady-state solution and it’s bifurca-tions in time-delay systems with band-limited feedback.This is a first step in a rigorous study concerning the ef-fects of AC-coupled components in nonlinear devices withtime-delayed feedback. We show that the steady state losesstability, generically, via a Hopf-bifurcation and we deter-mine whether the Hopf bifurcation is supercritical or sub-critical. Furthermore, the presence of double-Hopf bifurca-tions is shown, which indicates possible quasiperiodic andchaotic dynamics.

Daniel J. Gauthier, Lucas IllingDuke [email protected], [email protected]

MS42

New Directions in Nonlinear Delay OptoelectronicOscillators for Chaos Generation and Applicationto Wideband Communication Systems

Ikeda-like dynamics have been already intensively studiedfrom a fundamental point of view, both for their extremelysimple mathematical modeling (scalar nonlinear differencedifferential equation) together with their high complexitydynamical behavior. However, its experimental implemen-tation was often performed by optoelectronics equivalentsetups due to practical limitations. More recently, manysuch optoelectronics nonlinear delay dynamics have been

DS05 Abstracts 135

developed for wideband chaos communication applications.New dynamical models appeared with these optoelectronicoscillator.

Laurent LargerUniversite de Franche-ComteUMR FEMTO-ST [email protected]

Pierre-Ambroise Lacourt, Stephane Poinsot, AurelienPallavisiniGTL-CNRS TelecomUMR FEMTO-ST [email protected], [email protected], [email protected]

MS42

Dynamics of Chaotic Blocking Oscillators

Simple modification of a well-known circuit of blocking os-cillator can lead to the onset of chaotic oscillations. Theoutput signal of such chaotic oscillator is a series of short-term pulses characterized by chaotic fluctuations of timeintervals between pulses and has an Ultra-Wide Band con-tinuous power spectrum. We discuss the results of theo-retical and experimental studies of nonlinear dynamics ofthe oscillator. The bifurcation scenarios responsible for theonset and development of chaos are considered.

Nikolai RulkovUniv of California / San DiegoInst for Nonlinear [email protected]

Alexander VolkovskiiInst for Nonlinear ScienceUniv of California/San [email protected]

MS42

High-Frequency Implementation Approaches andCircuit Issues for PWL Chaotic Oscillators

The robust implementation of high-frequency chaotic oscil-lators is essential to the development and successful tech-nology insertion of many proposed chaos-based communi-cations and signal processing applications. This talk willtell the story of one journey towards arriving at such a ro-bust design, covering such topics as motivation for choos-ing PWL oscillators, the two basic dynamical system ap-proaches taken (autonomous and nonautonomous) for theoscillator implementation, and the various circuit issuesencountered and overcome along the way.

Christopher P. Silva, Albert YoungThe Aerospace [email protected], [email protected]

MS43

Smooth Dependence of Holonomy Maps of Hyper-bolic Systems

For a diffeomorphism possessing a hyperbolic attractor,the stable foliation induces a holonomy map between twosmooth submanifolds transversal to the foliation. Eventhough this holonomy map is only a Holder continuousmap, in general, the dependence of the holonomy map onthe diffeomorphism is differentiable when the transversal

smooth submanifolds are carefully chosen. Furthermore,the Jacobian of the holonomy map also depends differen-tiably on the diffeomorphism.

Miaohua JiangWake Forest UniversityDepartment of [email protected]

MS43

Billiards with Moving Walls

We consider a particle inside a billiard table with periodi-cally moving walls and show that under certain conditionsthe energy of some orbits approaches infinity at an affinerate. This behavior coexists with KAM tori of lower di-mension.

Mark LeviDepartment of MathematicsPennsylvania State [email protected]

MS43

Fixed Points for Commuting Diffeomorphisms onthe Sphere

Let G be a finitely-generated abelian group actingsmoothly on the two-dimensional sphere. Then there existsan index-two subgroup of G that acts with a global fixedpoint.

Kamlesh ParwaniUniversity of [email protected]

MS43

Nonuniformly Hyperbolic Dynamical Systems andPeriodic Orbits

Consider a smooth diffeomorphism on a compact Rieman-nian manifold preserving an ergodic hyperbolic measure ofpositive metric entropy. Assume that the metric entropy isnot locally maximal in the class of invariant ergodic hyper-bolic measures. We show that there exist multiplicativelymany periodic orbits equidistributed with respect to thismeasure.

Ilie UgarcoviciRice University, [email protected]

MS43

An Example of the CLT in the Absence of Mixing

The Central Limit Theorem is often obtained as a conse-quence of rapid mixing. However mixing is not required forthe CLT. We consider the CLT for regular observations onnon-mixing toral extensions of hyperbolic basic sets. Weconsider constant extensions and show that under appro-priate arithmetic assumptions on the extension and undermild mixing assumptions on the base we get a CLT.

Alistair WindsorUniversity of Texas, [email protected]

Ian Melbourne

136 DS05 Abstracts

University of Surrey, [email protected]

MS43

Recent Progress in Regularity of CohomologyEquations

We will present some new results on the problem of prov-ing regularity of solutions of cohomology equations over adynamical system. Most of the results discussed will be forsystems that have some partial or non-uniform hyperbol-icity, but we will discuss some parabolic systems.

Rafael de La LlaveUniversity of TexasDepartment of [email protected]

MS44

Persistent Gamma Oscillations and Their PossibleRole in Attention

We model persistent gamma oscillations, characterized invitro by infrequent, irregular, phase-locked spiking of ex-citatory cells. We hypothesize that similar oscillations invivo are a correlate of sustained attention. Subpopulation-specific excitation to E-cells drives them to spike at gammafrequency. Competition between excited subpopulationsrelies on acceleration (not strengthening) of the inhibitorypopulation rhythm. Model M-currents abolish rhythmicityand weaken response to specific excitation. Our findings fitwell with known effects of acetylcholine in cortex.

Nancy KopellBoston UniversityDepartment of [email protected]

Christoph BorgersTufts [email protected]

Steve EpsteinBoston UniversityCenter for [email protected]

MS44

Networks of Phase Oscillators: Beyond Symme-tries

Reduced models of oscillators are frequently used in mod-eling neuronal networks. Particularly popular are modelsin which the internal dynamics of the oscillators (cells) isone dimensional, such as integrate and fire, and phase os-cillators. We show that in networks of such oscillators thearchitecture (connectivity) can force the frequency of os-cillations of subsets of cells to be equal. A corollary of thisresult is that cells in an all-to-all coupled network of phaseoscillators all have the same frequency. These results aregeneralized to clusters of synchronous oscillators in net-works with arbitrary architecture using the ”groupoid” de-scription of a network’s structure. The relations betweenthe dynamics of the different oscillators and clusters aredescribed using the notions of rotation number, oscillationnumber, and average frequency. We also prove a specialcase of the H/K theorem for phase oscillators. This the-

orem and its corollaries restrict the possible spatial andspatiotemporal symmetries of periodic solutions in equiv-ariant phase oscillator networks and also guarantee thatperiodic solutions with certain symmetries will exist foropen sets of such systems. The network architecture againplays an essential role in determining which spatiotemporalsymmetries of the different solutions are permitted. Whilethe results are quite general, we illustrate them in networksof phase oscillators coupled through biophysically realisticphase response curves, and H functions.

Martin Golubitsky, Kresimir JosicUniversity of HoustonDepartment of [email protected], [email protected]

Eric Shea-BrownCourant Institute, New York Universitytba

MS44

The Dynamics of Cognition : Insights and Ques-tions from Human Direct Cortical Recordings

For the last few years, we have been recording the intracra-nial electroencephalogram of epileptic patients while theyperformed a variety of cognitive tasks, involving the per-ception of complex objects, language, verbal short-termmemory or the generation of occular saccades. The ex-ceptional spatial and temporal resolution of such record-ings gives a view of the brain dynamics in relation to cog-nition that is out of reach by conventional techniques ofhuman brain mapping. We will review some attempts tounderstand the organization of this dynamics along spa-tial,temporal and frequency axis and expose the needs forfurther tools of investigation adapted to this level of obser-vation.

Jean Philippe LachauxInserm, Lyon . [email protected]

MS44

Synchrony In Clinical Neurology

Using a new measure (Synchronization likelihood or SL)I studied normal and disturbed cognition states. Duringvarious types of cognitive tasks, SL reveals changes in spe-cific frequency bands, related to different aspects (workingmemory, attention) of information processing. In neuro-logical disorders with cognitive decline (Alzheimers andParkinsons disease) SL shows a decrease L in upper al-pha, beta and gamma bands. Magnetoencephalography isprobably more sensitive in detecting these changes thanEEG.

Cornelius StamVU University Medical Center, [email protected]

MS44

Where Noise and Precision Join Forces: Coding ofNeural Information Via Limit Cycles

In biological neural networks, the noise component oftenis of the same order as the signal strength. This, and the(as yet unexplained) computational efficacy of biologicalsystems, is of particular interest for technical exploitation,

DS05 Abstracts 137

since miniaturization drives hardware chips naturally to-wards these conditions. We discuss a framework in whichnoise and precision are complementarily used to encode in-formation. Our approach is based on weakly coupled neu-rodynamical limit-cycle solutions, which are investigatedunder natural conditions of transient temporal behavior.We have developed tools to show that locking is preservedunder a large variety of such conditions, although in thesecases the locked states are difficult to assess. We find thatthe range of conditions under which coding by locking is op-erational, is large enough for a realization in nature, hintingat a large potential also in technical applications. The de-scribed coding mechanism may be fundamental for achiev-ing the highly efficient computations observed in biologicalsystems.

Ruedi Stoop

Institute of Neuroinformatics ETHZ/[email protected]

MS44

Transient Stimulus-Locked Desynchronization andResponse Clustering in the Brain

Studies of transient responses of coupled oscillators to briefstimuli have led to new insights into the nature of evokedbrain responses: (i) Apart from generating a stereotypicalresponse, interacting oscillators may also switch betweendifferent responses across trials. This can typically notbe detected with standard averaging. (ii) The standardmethod for estimating transmission times fails when ap-plied to oscillators. Both predictions have been verified ina magnetoencephalograhy study with visual pattern rever-sal stimulation.


MS45

Stabilizing Unstable Waves

Chris JonesDepartment of MathematicsUniversity of North [email protected]

MS45

Homoclinic and Hopf Bifurcations in Calcium andFitzHugh-Nagumo Models

A numerical bifurcation analysis is performed of the in-teraction between a homoclinic and a Hopf bifurcationin models of Calcium wave propagation and a FitzHugh-Nagumo model. At first sight, the homoclinic orbit showssome unexpected behaviour when the corresponding equi-librium undergoes a Hopf bifurcation, but much can be ex-plained by the underlying slow manifold and the stiffnessof these slow-fast systems.

Bart E. OldemanDepartment of MathematicsUniversity of [email protected]

MS45

Bifurcations From Heteroclinic Networks with Pe-riodic Orbits

We consider bifurcations and bifurcation failure of hetero-clinic orbits from heteroclinic networks whose nodes involveperiodic orbits. A dual point of view is the termination of aparameter path at such a network. In some cases, these bi-furcations significantly differ from networks where all nodesare equilibria. We present results showing such differencesfor simple networks. A major application is the existence,and partially the stability, of travelling waves in spatiallyonedimensional parabolic PDEs.

Jens RademacherUniversity of British ColumbiaDepartment of [email protected]

MS45

Modeling Intracellular Calcium: Diffusion, Do-mains, and Dynamics

Mathematical models of calcium dynamics often representthe interaction of intracellular calcium with endogenousand exogenous calcium buffers via a system of nonlinearreaction-diffusion equations. For example, emisphericallysymmetric steady-state solutions to the full equations forthe buffered diffusion of intracellular calcium provide esti-mates of the concentration of free calcium near open cal-cium channels. Perturbation methods have provided ap-proximations for steady-state calcium and buffer profilesin two well-understood asymptotic limits. 1) An ”excessbuffer approximation” (EBA) where the mobility of bufferexceeds that of calcium and the fast diffusion of buffer to-ward the calcium channel prevents buffer saturation (Ne-her, 1986; Naraghi and Neher, 1997). 2) A ”rapid bufferapproximation” (RBA), where the diffusive time scale forcalcium and buffer are comparable, but slow compared toreaction, resulting in saturation of buffer near the calciumchannel (cf. Wagner and Keizer, 1994; Smith, 1996). Thesetwo limits can also be applied to investigations of the re-lationship between single-channel kinetics and the collec-tive phenomena of stochastic Ca2+ excitability, i.e., localcalcium elevations known as calcium ”puffs” and ”sparks.”The interaction of calcium and buffers also plays an impor-tant role in the dynamics of both continuous and saltatorypropagating calcium waves.

Gregory D. SmithCollege of William and [email protected]

MS46

Multiscale Simulation of Sedimentation

Sedimentation at small Reynolds numbers is a simple-to-define process at the microscale level of hydrodynamic in-teractions between solid particles, but full simulations arecomputationally intensive. Meanwhile, there is no clearagreement on the precise form of the correct macroscopiccontinuum models, even in the dilute limit of monodispersespheres. In this talk, we will demonstrate attempts to useequation-free methodologies to efficiently capture the richbehavior of the statistics of particle velocities and the evo-lution of the spreading front between particle-laden andclarified flow.

Peter J. Mucha

138 DS05 Abstracts

Georgia Institute of TechnologySchool of [email protected]

MS46

Application of Equation-Free Methods to Modelsof Biological Systems

The movement of many organisms can be described as arandom walk at either or both the individual and pop-ulation level. The rules for this random walk are basedon complex biological processes and it may be difficult todevelop a tractable quantitatively-accurate individual-levelmodel. However, important problems in areas ranging fromecology to medicine involve large collections of individuals,and a further intellectual challenge is to model population-level behavior based on a detailed individual-level model.Because of the large number of interacting individuals andbecause the individual-level model is complex, classical di-rect Monte Carlo simulations can be very slow, and often oflittle practical use. In this case, newly developed equation-free methods may prove a promising direction for analysisand simulation of individual-based models. We will discussapplications of equation-free methods to biological prob-lems. The biological example here is chemotaxis, but itcould be any random walker which biases its movement inresponse to environmental cues.

Hans OthmerUniversity of MinnesotaDepartment of [email protected]

Radek ErbanSchool of MathematicsUniversity of [email protected]

Ioannis KevrekidisPrinceton [email protected]

MS46

Higher Order Accuracy in The Gap-Tooth SchemeFor Large-Scale Solutions Using Microscopic Sim-ulators

We find general boundary conditions for patches of micro-scopic simulators that appropriately connect widely sep-arated “teeth” to achieve high order accuracy over themacroscale. Here we explore the simplest case when the mi-croscopic simulator is the quintessential example of a par-tial differential equation. We argue that classic high-orderinterpolation provides patch boundary conditions whichachieve arbitrarily high-order consistency in the gap-toothscheme, and with care are numerically stable. This consis-tency is demonstrated: firstly using the dynamical systemsapproach of holistic discretisation; and secondly throughthe eigenvalues of selected numerical problems.

Tony RobertsUniversity of Southern [email protected]


MS46

Patch Dynamics With Buffers For Multiscale Prob-lems

For an important class of multiscale problems, a separa-tion of scales exists between the available (microscopic)model and the (macroscopic) level at which one would liketo observe the system. For this problenm, Kevrekidis etal. developed a so-called “equation-free” framework, ofwhich patch dynamics is an essential component. Patchdynamics is an algorithm designed to approximate the timeevolution of the macroscopic unknowns; it only performsappropriately initialized simulations with the available mi-croscopic model in small portions of the space-time domain(the patches). To obtain convergence, we introduce bufferregions around the patches to avoid the (artificially intro-duced) patch boundaries to affect the solution in the in-ternal region. We show our recent convergence results andillustrate the approach by on a set of model problems.

Giovanni SamaeyDepartment of Computer Science, K. U. [email protected]

Dirk RooseK.U.LeuvenDept. of Computer [email protected]


MS47

Curve Shortening and the Topology of ClosedGeodesics on Surfaces

Sigurd B. AngenentUniversity of WisconsinDepartment of [email protected]

MS47

Constructing Models of Global Attractors forSwift-Hohenberg

In this talk, we will discuss a rigorous numerical method forthe study and verification of global dynamics for gradientsystems. The procedure involved relies on first verifyingthe structure of the set of stationary solutions, as oftendepicted in a bifurcation diagram produced via continu-ation methods. This includes proving the existence anduniqueness of computed branches of the diagram as wellas showing the nonexistence of additional stationary solu-tions. Topological information in the form of the Conleyindex, also computed during this verification procedure, isthen used to build a model for the attractor. As illustra-tion, we apply this method to the Swift-Hohenberg PDEto produce a conjugacy between the global attractor a con-structed model system.

Sarah DayDepartment of MathematicsCornell [email protected]

DS05 Abstracts 139

MS47

Structure of the Attractor of the Cahn-HilliardEquation

Abstract: We describe the structure of the attractor of theCahn-Hilliard equation on two-dimensional square domainsin certain parameter ranges. This is accomplished by com-bining numerical results on the set of equilibrium solutionswith algebraic Conley index techniques such as connectionmatrices and transition matrices. In particular, we discussthe possibility of saddle-saddle connections.

Stanislaus Maier-PaapeRWTH AachenInstitut fur [email protected]

MS47

Braids and Parabolic Equations

The comparison principle for scalar second order parabolicPDEs admits a topological interpretation: after lifting thegraphs to Legendrian braids, the curves evolve as to de-crease the algebraic length of the braid. Via discretizationwe define a suitable Conley index, which gives a toolboxof purely topological methods for finding invariant sets ofscalar parabolic PDEs. There is a close connection withtwist maps and in this context it applies to (variational)fourth order ODEs.

Robert VandervorstVU AmsterdamDepartment of [email protected]

Robert W. GhristDepartment of MathematicsUniversity of Illinois, [email protected]

Jan Bouwe Van Den BergVU AmsterdamDepartment of [email protected]

MS48

Renormalization for Critical Shearless Circles

We discuss the numerical evidence for universal rescalinginvariance of shearless critical circles of nontwist maps withnoble rotation number. It is also seen that the nearby pe-riodic orbits map onto each under simultaneous rescalingand shift of rotation numbers. We present a renormaliza-tion group (RG) picture to interpret these observations interms of critical fixed points (or higher period cycles) of theRG operators acting on the space of maps. We also con-struct changes of coordinate to relate different fixed pointsto each other.

Amit ApteDept. of MathematicsUniversity of North [email protected]@email.unc.edu

Philip MorrisonPhysics DepartmentUniversity of Texas at [email protected]

Alex WurmDept. of Physics, Fusion StudiesUniversity of Texas at [email protected]

MS48

Destruction of Invariant Circles and TransportBarrier Location in the Dynamics of QuadraticNontwist Maps

We study the scenario of destruction of invariant circles ofquadratic nontwist maps. It is proved that the main causeof their breakup is the folding property of the map. For themaps corresponding to a point in a distinguished subregionof the parametric plane we locate a closed invariant annu-lus and show that outside it the drifting of action occurs.Theoretical results are illustrated for a global model of aPoincare map associated to a poloidal section of a reversedshear tokamak.

Emilia PetrisorDepartment of MathematicsPolitechnic University of [email protected]

Dana ConstantinescuEURATOM MEC [email protected]

Jacques MisguichEURATOM CEA [email protected]

MS48

Regularity of Critical Invariant Circles of Non-Twist Maps

We study critical invariant circles of several noble rota-tion numbers at the edge of breakdown for area preservingmaps of the cylinder which violate the twist conditions. Wepresent a high accuracy computation of about 10 millionFourier coefficients. This allows us to compute the regular-ity of the conjugating maps and show that, to the extent ofthe precision, it only depends on the tail of the continuedfraction expansion.



Nikola PetrovUniversity of [email protected]

MS48

The Diagram for Shearless Torus Breakup and Sep-aratrix Reconnection in the Quadratic NontwistMap

The appearance of a shearless torus and the reconnection ofseparatrices are phenomena peculiar to nontwist systems.For the quadratic nontwist map, these can be systemati-

140 DS05 Abstracts

cally studied by investigating the trajectories of four par-ticular points in phase space that are the fixed points oftransformations related to the map’s symmetry. By us-ing these points, a diagram is obtained which reveals thebreakup threshold of shearless tori and the reconnectionthreshold of separatrices.

Susumu ShinoharaATR Wave Engineering [email protected]

MS49

The Dynamics of Behavioral Choice and its Effectson Population Dynamics

I will extend previous work on the dynamics of behavioralchange to consider systems in which several interactingspecies adaptively adjust behavioral traits that affect theirinteraction. A range of different behavioral models is an-alyzed. All of the models have the characteristic that theaverage value of the behavioral trait in each species changesmore rapidly when the gain in fitness per unit change inthe behavior is larger. The work will examine the popula-tion dynamics of the entire system, and will explore howthese dynamics are affected by: (1) the type of interactionbetween species; (2) the functional relationship betweentraits in different species and population growth parame-ters; and (3) the precise form of the relationship betweenthe fitness gradient and the rate at which behavior changes.

Peter AbramsDept. of Zoology, University of [email protected]

MS49

Adaptation Model of Phytoplankton Growth FarFrom Equilibrium

We investigate the growth of a phytoplankton populationin a chemostat, where the initial state is perturbed far fromequilibrium. Experimental results are presented showinga large range of dynamics regimes, including strong over-shooting, sustained oscillations and non-intuitive responseto nutrient jumps. To describe the experimental findingswe develop a simple chaotic adaptation model taking intoaccount an internal phytoplankton state. Our results giveinsights into the dynamics of nutrient limited growth farfrom equilibrium.

Bernd BlasiusDept. of PhysicsUniversity of [email protected]

MS49

Adaptive Dynamics in General Models

In general models the functions that specify the interactionof model variables are not restricted to specific functionalforms. In this way a single general model describes a largeclass of different systems. Despite this generality, generalmodels can be analysed efficiently in the framework of lo-cal bifurcation theory. In this talk we apply the generalmodelling approach to adaptive dynamical processes. Weillustrate the formulation and investigation of general mod-els of adaptive dynamics and discuss their advantages.

Ulrike Feudel

University of OldenburgICBM, Theoretical Physics/Complex [email protected]


Dirk StiefsICBM, Theoretical Physics/Complex SystemsCarl von Ossietzky University Oldenburg, [email protected]

MS49

Evolutionary Dynamics of Mixotrophs

We study the evolution of a population of mixotrophs livingin a water column, where light absorption triggers the buildup of concentration gradients. The mathematical modelconsists of an ecological part that describes the dynamicsof the population and its nutrients with partial differentialequations (PDEs) and an evolutionary part that describesthe change of strategies (autotroph and heterotroph abil-ities) of these populations, parameters in the PDEs, onthe slower time-scale.In this approach competition betweenthe resident population and a mutant population deter-mines the evolutionary dynamics.Under certain conditionsmutants and residents can coexist leading to evolutionarybranching, i.e. speciation.

Bob KooiDept. of Theoretical Biology, Vrije Universiteit,AmsterdamThe [email protected]

Tineke Troost, Sebastiaan KooijmanDept. Theoretical Biology, Vrije Universiteit AmsterdamThe [email protected], [email protected]

MS50

On the Turbulent Diffusion by Waves

Turbulent diffusion of a passive tracer by a random wavefield is believed to be quadratic with respect to the en-ergy spectrum εk of the velocity field (i.e. proportional toε4, where ε is the order of the wave amplitudes). So, thewave turbulence diffusion (say, on the ocean surface or inthe air) is often believed to be dominated by the turbulentdiffusion of the incompressible flow. In this presentation,we consider a different mechanism of the wave turbulentdiffusion and find that the wave turbulent diffusion canbe more significant than previously thought. This mecha-nism works if the velocity field is compressible and statisti-cally anisotropic, so that the wave system has a significantStokes drift. The contribution of this mechanism has alower order in ε. We confirm our results with numericalsimulations. To derive these results, we develop the Sta-tistical near-identity transformation. We also show that asmall deviation of the energy spectrum εk can lead to adrastic difference in the coefficient of turbulent diffusion.

Aleksander BalkThe University of [email protected]

DS05 Abstracts 141

MS50

Statistical Physics of Waves

David CaiCourant instituteNew York [email protected]

MS50

Anomalous Probability of Large Amplitudes inWave Turbulence

Brief overview of wave turbulence formalism will be pre-sented. Generalized Random-Phase-and-Amplitude ap-proach will be introduced. This approach will be usedto derive evolution equations for spectrum and ProbabilityDistributions Function of waves in a self consistent manner.It will be shown that stationary state of wave-turbulencesystems may correspond to intermittency (high probabilityof waves with large amplitudes).

Yuri V. LvovRensselaer Polytechnic [email protected]

MS50

Numerical Modeling of Surface Gravity Waves:Discrete Resonances

Gravity surface gravity waves has been traditionally in thefocal point of wave turbulence theory. Recenlty it becamepossible to model numerically underlying evolution equa-tions of surface waves. Spectral energy density of gravitywaves will be measured and compared to measurements ofother groups and predictions of WT. Significant additionalinformation will be measured as well, including spectralfluctuations, phase statistics and probability disctributionfunctions of waves. Difficulties of numerical modelling,associated with discreteness of spectral space will be dis-cussed. It will be demonstrated that statistical steady stateof gravity surface waves correspond to strong intermittencyfor low wave-numbers, due to the flux of probability in anamplitude space.

Boris PokorniRensselaer Polytechnic [email protected]

MS51

The Interplay of Mixing, Reaction, and Diffusion:Experiments and Simulations

We investigate a fast acid-base reaction in the presenceof chaotic advection and diffusion, in a thin layer. Wecompare experimental results to theory and simulations,which indicate that after a short transient, the productconcentration should increase exponentially at first due tostretching, and then more slowly due to diffusion and thecollision of reaction interfaces. We demonstrate experimen-tally the close connection between reaction and stretching.However, the reaction product grows more slowly than ex-pected, possibly as a result of highly non-uniform stretch-ing. Surprising oscillations occur on a time-scale muchslower than the basic flow period, a phenomenon that isnot reproduced by the usual model for fast reactions, whichis isomorphic to a passive scalar problem. We discuss pos-

sible origins of the unexpected oscillations.

Zoltan NeufeldUniversity College [email protected]

Jerry Gollub, Paulo ArratiaHaverford [email protected], [email protected]

MS51

Recent Mathematical Advances on Strange Eigen-modes

Diffusive tracers tend to develop intriguing patterns whenmixed by unsteady fluid flows. These complex patterns, orstrange eigenmodes, appear in a broad class of flows, rang-ing from simple laboratory mixing experiments to simu-lations of atmospheric dynamics. Despite their ubiquity,the patterns have not been understood mathematically.In this talk, I discuss recent analytic results on the ex-istence and evolution of strange eigenmodes in two andthree-dimensional fluid flows. I also describe similar re-sults for the wilde eigenmodes observed in dynamo theory.


MS51

Chaotic Advection-Diffusion Problems For a TracerSubject to Phase Transitions

We consider the planar advection-diffusion problem for ascalar tracer subject to a phase change. An importantexample of this class of problem is condensation of wa-ter vapor in the Earth’s atmosphere. The target quantityfor understanding in that case is the probability distribu-tion of subsaturation, a quantity that has profound im-plications for the climate of the Earth. The condensationproblem is a prototype for a broad class of problems in-volving nonlinear chemical reactions. We point out that,because condensation is a nonlinear process, the operationsof coarse-graining concentrations on a cloud of advectedparticles does not commute with the operation of conden-sation. This means that diffusion cannot be gotten intothe problem using the familiar trick of adding a random-walk component to deterministic trajectories. Some con-sequences of this are pointed out. A number of numericalexplorations mapping out the behavior of the system arepresented.

Raymond T. PierrehumbertThe University of ChicagoDept. of the Geophysical [email protected]

MS51

Topological Kinematics of Mixing

The orbits of fluid particles in two dimensions effectivelyact as topological obstacles to material lines. A spacetimeplot of the orbits of such particles is a braid whose prop-erties reflect the underlying dynamics. For a chaotic flow,the braid generated by the motion of three or more fluidparticles is computed. A “braiding exponent” is defined tocharacterize the complexity of the braid. This exponent is

142 DS05 Abstracts

proportional to the usual Lyapunov exponent of the flow.Measuring chaos and mixing properties in this manner hasseveral advantages, since neither nearby trajectories norderivatives of the velocity field are needed.

Matthew Finn, Jean-Luc ThiffeaultImperial College [email protected], [email protected]

MS52

Stimulus-Induced Traveling Waves and Breathersin Neural Networks

We analyze the existence and stability of stimulus-lockedstationary (in 1-D and 2-D) and traveling (in 1-D) pulsesolutions in synaptically coupled neural networks. The net-works are modeled in terms of nonlocal integrodifferentialequations, in which the integral kernal represents the spa-tial distribution of synaptic weights and the output firingrate of a neuron is taken to be a Heaviside function of activ-ity. We explore the emergence of time-periodic pulse-likesolutions, or breathers, which arise as the system under-goes a Hopf bifurcation, rendering the pulse unstable.

Stefanos FoliasUniversity of [email protected]

MS52

Heterogeneous Connections Cause StabilityChanges in Neural Field Propagation

Spatiotemporal pattern formation in physical and chemicalsystems is typically based on a dynamics with a homoge-neous, i.e. translationally invariant, connection topology.However, biological systems like the human cortex showhomogeneous connectivity with additional strongly hetero-geneous projections from one area to another. Here wereport how such a dynamic system performs macroscopi-cally coherent pattern formation and provide a formalismfor its analytic treatment. The connection topology is usedsystematically as a control parameter to guide the systemthrough a series of phase transitions. We discuss the exam-ple of a two-point connection and its destabilization mech-anism.

Viktor JirsaFlorida Atlantic UniversityPhysics [email protected]

MS52

Periodic Travelling Waves In the Theta Model ForSynaptically Connected Neurons

We present mathematical results obtained about periodictravelling waves in the theta model for a linear continuumof synaptically-interacting neurons. In the case of excitableneurons, we prove that periodic travelling waves exist whenthe synaptic coupling is sufficiently strong, and in that caseat least two periodic travelling waves of each wave-number,a ‘fast’ and a ‘slow’ one, exist. We also study the limitsof large wave-number and of small wave-number, for whichmore information can be obtained. Some numerical results,and open questions, will also be presented.

Haggai KatrielEinstein Institute of MathematicsThe Hebrew University

[email protected]

MS52

Synaptic Saturation-Induced Effects on Dynamicsof Traveling Waves in Integrate-And-fire NeuralNetworks

We extend upon the previous theoretical and computa-tional studies of traveling waves of activity in neural tis-sue by analyzing how saturated synapses change the dy-namics of interacting spiking neurons. The addition ofbiologically-motivated pre and post-saturated synapses tothe traditional models has the desired property of prevent-ing potential explosion of self-generated network activity.Asymptotic analytical solutions are compared to dynam-ics generated by numerical simulations for one and two-dimensional integrate-and-fire neural networks.

Remus OsanBoston [email protected]

MS53

Dynamical Systems Approaches to Light Propaga-tion in Optical Fibers

John AbbottCorning [email protected]

MS53

An Epidemiological Model of Alcohol ProblemTreatment

A model of alcohol-related problem treatment on a socialnetwork is developed. A discrete model using a cubic non-linearity is used on both random and rewired-connected-caveman network. The model modifies the likelihood todevelop an alcohol based on a comparison with the aver-age of the values at the neighboring vertices. A treatmentmodel is introduced that resets the value at the vertex andholds it constant over a number of time steps. Significantdependence on the treatement parameters is found.

Richard BraunUniversity of Delaware, Newark, DEDepartment of Mathematical [email protected]

MS53

Modeling and Simulation of High-Speed MachiningOperations

The most basic manufacturing processes involve the me-chanical working of a material, resulting in a permanentalteration of its shape to produce a finished component.Many industrial organizations still use trial-and-error pro-totyping to select process parameters. This method isexpensive, and it often leads to sub-optimal parameterchoices. A necessary step towards improved process controlis the development of better models of these operations.While considerable progress has been made in the devel-opment of predictive models for low-strain-rate processes,there is currently a need for improved predictive capabili-ties for high-rate processes. In this talk, a survey will begiven of some work in progress at NIST on the the mod-eling and simulation of some basic high-speed machining

DS05 Abstracts 143

operations.

Timothy J. BurnsNational Institute of Standards and [email protected]

MS53

Control and Connectivity of Networked DynamicalSystems

In this work we consider networked dynamical systems con-trol dependency on available communication links. UsingGraph-theoretic tools we focus on designing networked dy-namical systems decentralized control law. We model over-all system in a way that allows for both fixed and varyingcommunication topology. In the case of variable communi-cations topology, communication links between individualsystems are defined based on a predefined proximity rule.Dynamical systems considered are homogeneous, with dis-crete time dynamics.

Anca WilliamsPortland State [email protected]

Sonja [email protected]

MS54

Nonautonomous Dynamics of Geophysical Flows

Jinqiao DuanIllinois Institute of [email protected]

MS54

Some Questions In Control of Nonautonomous Sys-tems

Using methods of the theory of nonautonomous lin-ear differential systems we generalize the statement ofYakubovich’s Frequency Theorem from periodic controlsystems to systems with bounded uniformly continuouscoefficients. Then we study nonautonomous H∞ controlproblmes with infinite horizon. We pass from a Riccatiequation to a linear nonautonomous Hamiltonian system.Using the concepts of exponential dichotomy and rotationnumber we define a minimal attenuation value and provestability when the disturbance is zero.

Russell JohnsonUniversita di [email protected]

Roberta FabbriUniversity of [email protected]

Carmen NunezUniversity of [email protected]

MS54

Set Oriented Methods for Optimal Control Prob-lems

We develop a set oriented method for approximating theoptimal value function and approximate optimal trajecto-ries of nonlinear optimal control problems. The idea of themethod is to employ a set oriented approach in order to ex-plicitly construct a weighted directed graph that is a finitestate model of the original continuous space control sys-tem. On this graph, standard graph theoretic algorithmsfor computing (all source, single destination) shortest pathscan be applied in order to compute an approximate optimalvalue function. The associated feedback law gives rise toapproximate optimal trajectories and we derive statementsabout the performance of this feedback. We also show howto generalize this approach to perturbed systems and inparticular, how robust feedback laws can be constructed.



Lars GruneMathematical InstituteUniversitaet [email protected]

MS54

Inertial Manifolds For Nonautonomous Skew Prod-uct Semiflows and Applications

Under the dissipative conditions, the mild solutions of anonlinear nonautonomous evolutionary equation du/dt +Au = F(u,t) can be formulated as a skew product semiflowin a product phase space. We show that there exists aninertial manifold for this skew product semiflow under thespectral gap condition. Instead of using Lyapunov-Perronmethod, we take the approach of conic invariance and in-cremental exponential dichotomy to fulfil the proof, basedon two conic differential inequalities. The construction ofinertial manifold is through an exponentially tracking in-tegral manifold with the aid of Caccioppoli homotopy. Anillustration is made by nonautonomous reaction-diffusionequations.

Yuncheng YouUniversity of South [email protected]

MS55

Patterns of Synchrony in Lattice Dynamical Sys-tems

Stewart et al. have shown that flow invariant subspaces forcoupled networks are equivalent to a combinatorial notionof a balanced coloring. Wang and Golubitsky have classi-fied all balanced two colorings of planar lattices with eithernearest neighbor (NN) or both nearest neighbor and nextnearest neighbor coupling (NNN). This classification givesa rich set of patterns and shows the existence of manynonspatially periodic patterns in the NN case. However,all balanced two-colorings in the NNN case on the squareand hexagonal lattices are spatially periodic. We present

144 DS05 Abstracts

new results showing that all balanced k-colorings in theNNN case on square, rhombic and hexagonal lattices arespatially periodic.

Ana Paula S. DiasUniversity of [email protected]

Yunjiao WangUniversity of [email protected]

Fernando M. AntoneliUniversity of San [email protected]

Martin GolubitskyUniversity of HoustonDepartment of [email protected]

MS55

Synchrony-Breaking Bifurcations in Coupled CellSystems

Network architecture can lead to robust synchrony in cou-pled systems and surprisingly to codimension one nilpo-tent normal forms in Jacobians associated to synchronousequilibria. We analyse codimension one nilpotent Hopfbifurcations that occur generically in three different net-works. Phenomena stemming from these bifurcations in-clude multiple periodic solutions, solutions whose growthrate is faster than the standard 1/2 power, and solutionswhose growth rate is slower than 1/2 power.

Toby ElmhirstUniversity of [email protected]


MS55

Some Examples of Coupled Cell System Dynamics

We consider examples of coupled cell networks with syn-chronous dynamics that are unexpected from symmetryconsiderations but are natural using a theory developed byStewart, Golubitsky, and Pivato. Our examples includepatterns of synchrony in networks with small numbers ofcells and in lattices (and periodic arrays) of cells that can-not readily be explained by conventional symmetry consid-erations. The examples we consider include a 3-cell systemexhibiting equilibria, periodic, and quasiperiodic states indifferent cells; periodic 2n × 2n arrays of cells that gen-erate 2n different random patterns of synchrony from onesymmetry generated solution; and systems exhibiting mul-tirhythms (periodic solutions with rationally related peri-ods in different cells). Some of these phenomena will bediscussed in greater detail in other talks in this minisym-posia.

Matthew NicolUniversity of [email protected]

MS55

Bursting in Coupled Cell Systems

Periodic bursting in fast-slow systems can be viewed asclosed paths through the unfolding parameters of degen-erate singularities. Using this approach, we show thatbursting in coupled systems can have interesting behav-ior. We show that coupled systems with Z2 symmetry,particularly in Hopf/Hopf mode interaction and symmetry-breaking Takens-Bodganov singularities, lead to interestingbursting phenomena.

LieJune ShiauUniversity of Houston - Clear LakeDepartment of [email protected]

Martin Golubitsky, Kresimir JosicUniversity of HoustonDepartment of [email protected], [email protected]

MS56

Stochastic Epidemic Models For the Spread of Han-tavirus in Rodents

Hantavirus infection is an emerging disease carried by ro-dents. In rodents, hantavirus infection has little impactbut in humans, infection is manifested as hantavirus pul-monary syndrome with a mortality rate as high as 50%. Wepresent some deterministic and stochastic epidemic modelsfor the spread of hantavirus in rodents.

Linda AllenDept. of Mathematics and StatisticsTexas Tech [email protected]

MS56

TBA

Julien ArinoDepartment of MathematicsMcMaster [email protected]

MS56

TBA

Carlos Castillo-ChavezDepartment of Mathematics and StatisticsArizona State [email protected]

MS56

The Spread of Epidemics: Hantavirus and WestNile Virus As Examples

Nitant KenkreDepartment of PhysicsUniversity of New [email protected]

DS05 Abstracts 145

MS56

Non-Local Dispersal and the Spatial Spread of Dis-ease

Jan MedlockUniversity of WashingtonApplied [email protected]

MS57

On the Campbell-Sigeti Conjecture

The Campbell-Sigeti conjecture for dissipative dynamicalsystems says that the sum of positive Lyapunov exponentsis proportional to the width of the band of analyticity (incomplexified time) of the solutions on the global attractor.We outline a partial analytical result towards solving thisconjecture and test it computationally for the Kuramoto-Sivashinsky equation.

Alexey CheskidovDepartment of MathematicsUniversity of [email protected]

MS57

Using Svd to Approximate Spectral Intervals ofContinuous Dynamical Systems

We explore numerical techniques based on the continuousSingular Value Decomposition to approximate the Lya-punov spectrum and the exponential dichotomy spectrumof n-th dimensional systems of ordinary differential equa-tions. Theoretical justifications and numerical examplesare given.

Cinzia EliaUniv. of Bari, [email protected]

Luca DieciSchool of MathematicsGeorgia [email protected]

MS57

Computation of Lyapunov Exponents for Dissipa-tive Systems

We compute Lyapunov exponents for both the Kuramoto-Sivashinsky and 2-D Navier-Stokes equations. A varietyof continuous and discrete QR methods are compared aswell as some which incorporate (for the K-S equation) analgorithm to compute inertial manifolds.

Luca DieciSchool of MathematicsGeorgia [email protected]

Mike JollyDepartment of MathematicsIndiana [email protected]

Erik Van VleckDepartment of Mathematics

University of [email protected]

MS57

The Evans Function and Fredholm determinants

For quite general nonautonomous first order systems of dif-ferential equations, we define the Evans function using Bohland Lyapunov exponents, and prove that the Evans func-tion is equal to the modified Fredholm determinant of anassociated integral operator with semi-separable kernel.

Yuri Latushkin, Fritz GesztesyDepartment of MathematicsUniversity of [email protected], [email protected]

Konstantin MakarovDepartment of MathematicsUniversity of [email protected]

MS57

Nonlocal Models for Directed Self-Assembly ofNano-Particles

We derive a set of non-local evolution equations describ-ing directed self-assembly of nano-particles due to mutualattraction. We show that our models exhibit stable gener-alized solutions as a set of moving delta-peaks (clumpons) ,and there is an exact finite-dimensional dynamics describ-ing the peaks. We also show that our methods can besuccessfully applied in some problems of crystal growth,eliminating the need for surface-tension like stabilizationin models for these phenomena.

Darryl D. HolmLos Alamos and Imperial [email protected]

Vakhtang PutkaradzeU [email protected]

MS57

An Error Analysis for the Approximation of Lya-punov Exponents

We present an error analysis for approximating Lyapunovexponents using QR techniques. The bounds derived area function of the LOCAL error in approximating the timedependent orthogonal change of variables Q and the de-gree to which there is integral separation. The stability ofLyapunov exponents with respect to perturbation is closelyrelated to integral separation.

Erik Van VleckDepartment of MathematicsUniversity of [email protected]

MS58

Mixed Mode Oscillations For a Chemical Oscillator

One known mechanism of formation of mixed mode oscil-lations is due to the presence of a folded node or a folded

146 DS05 Abstracts

saddle node combined with a suitable return mechanism.The mixed mode trajectories arising in this scenario area combination of rotations about the fold line and largeexcursions of relaxation type. Simple mixed mode oscil-lations (many small oscillations followed by one large) arequite robust, but solutions invloving more than one con-sequitive large oscillation are much harder to find. In thistalk we present ways of finding mixed mode oscillationsbased on the theoretical results developed in colaborationwith Krupa and Wechselberger. The context for our studyis given by a chemical oscillator considered by Jeff Moehlis(J. Nonl. Sci. Vol. 12 : pp. 319-345 (2002).

Morten BronsTech University of DenmarkDepartment of [email protected]

Maciej KrupaNew Mexico State UniversityDept. of Mathematical [email protected]

MS58

Bifurcation of Neural Models: Multiple TimeScales and Canards

Singular perturbation analysis of models for bursting neu-ral rhythms relies upon decomposition of models into slowand fast time scales. Transitions between spiking and qui-escent dynamics correspond to bifurcations of the fast sub-systems. We observe that the fast subsystems of burst-ing models also have two time scales. Consequently, thebifurcation analysis of the fast subsystems is surprisinglycomplex and involves canards. This talk illustrates thesephenomena with several examples.

John GuckenheimerCornell [email protected]

MS58

Mechanism of Mixed Mode Oscillations Via FoldedNode Points

Folded node points are a special type of fold points oc-curing in singularly perturbed systems with at least twoslow variables. Typical trajectories passing through theneighborhood of a folded node point make some numberof oscillations about the fold line and then follow the re-laxation mechanism. Two limits of folded node are foldedsaddle nodes of type I and type II, both characterized bythe number of oscillations increasing indefinitely. All thethree singularities combined with a suitable return mech-anism give rise to mixed mode dynamics occurring as acombination of small rotations and large excursions of re-laxation type. We present some results on mixed modeoscillations occurring in the context of folded node and itslimits. The analysis is based on geometric singular pertur-bation and the blow-up method.

Morten BronsTech University of DenmarkDepartment of [email protected]

Maciej KrupaNew Mexico State UniversityDept. of Mathematical Sciences

[email protected]

Martin WechselbergerThe Ohio State [email protected]

MS58

Mixed-Mode Oscillations in a Model of Solid Com-bustion

We study a system of three differential equations modelingsolid combustion. In the regime near an Andronov-Hopfbifurcation, the system exhibits a variety of oscillatory be-havior. The structure and bifurcations of stable periodicsolutions are investigated in this work.

Philip HolmesProg. in Applied and Comp. MathematicsPrinceton [email protected]

Georgi MedvedevDrexel UniversityPhiladelphia, [email protected]

Yun YooDepartment of MathematicsDrexel [email protected]

MS58

Panel Discussion



MS58

Reduced Systems and Canards for Neuron Models

We study the dynamics of coupled neurons modeled as sin-gularly perturbed systems of differential equations. Thefast/slow decomposition of such systems induces a reducedsystem that is often simpler to analyze than the full sys-tem. In order for the reduced system to capture the possi-ble behaviors and bifurcations of the full system, the phe-nomenon of canards must be incorporated in the reducedmodel. We will illustrate this process with examples, in-cluding a model of two coupled neurons.

Warren WeckesserColgate UniversityDepartment of [email protected]

Kathleen A. HoffmanUniversity of Maryland, Balt. Co.Deapartment of Math. and [email protected]

DS05 Abstracts 147


MS59

Orbital Dynamics in Extended Mass Distributions

In this talk I shall discuss aspects of orbital dynamics inan extended mass distribution. The main focus will be ona particular form of the potential that arises from the uni-versal form for the density profile of dark matter halos ascomputed via N-body simulations. I will consider stabilityof orbits, analogs of Kepler’s laws and the nature of tidalforcing during halo mergers. This is joint work with FredAdams, Michael Busha and Gus Evrard.

Fred AdamsDepartment of PhysicsUniversity of [email protected]

Anthony M. BlochUniversity of MichiganDepartment of [email protected]

MS59

Shape Dynamics and Control

A long thread of investigations in geometric mechanics,chemistry, and statistics has contributed to our current un-derstanding of shape spaces. In this talk, we explore somedynamical systems on shape spaces and associated controlproblems. This work is in part motivated by results ofa collaboration with Fumin Zhang and Eric Justh on thesubject of interacting particle systems.

P.S. KrishnaprasadUniversity of Maryland, College ParkInstitute for Systems [email protected]

MS59

Break-Up of Invariant Curves By Magnetic Field

We consider the motion of charged particles in the planewith a periodic magnetic field perpendicular to the plane.If the magnetic field has zero average, KAM theory givesquasiperiodic motions (under some additional assump-tions). Any change of the average of magnetic field fromzero destroys these tori. We study the structure of the re-maining set, as well as the physical manifestation of thestructure of this set.

Mark LeviDepartment of MathematicsPennsylvania State [email protected]

MS59

Variational Principles, Implicit Lagrangian Sys-tems and Dirac Structures

Implicit Lagrangian systems are defined and some of theirbasic properties developed in the context of variationalprinciples and Dirac structures. These systems includeconstrained systems, as well as networks of Lagrangianmechanical systems. Implicit Lagrangian systems are de-

fined by employing natural symplectomorphisms betweeniterated tangent and cotangent spaces. Variational princi-ples in an extended form (the Hamilton-Pontryagin prin-ciple) and how they are related to Dirac structures playan important role in the development. Employing theLagrange-dAlembert principle, mechanical systems withexternal forces as well as with nonholonomic constraintsare put into the context of implicit Lagrangian systems.Degenerate Lagrangian systems are included in the theoryof implicit Lagrangian systems and will be illustrated usingL-C circuits.


MS59

Rigid Multi-Frequency N-Vortex ConfigurationsOn the Rotating Sphere

The problem of N-point vortices moving on a rotating unitsphere is described. Through a sequence of linear coordi-nate transformations, we show how to reduce the problemto that on a non-rotating sphere, where the center of vor-ticity vector is aligned with the z-axis. As a consequence,we prove that integrability on the rotating sphere is thesame as on the non-rotating sphere. Rigid multi-frequencyconfigurations that retain their shape while rotating abouttwo independent axes with two independent frequenciesare obtained, and necessary conditions for one-frequencyand two-frequency motion are derived. Examples includingdipoles which exhibit global ‘wobbling’ and ‘tumbling’ dy-namics, rings, and Platonic solid configurations are shownto undergo either periodic or quasi-periodic evolution onthe rotating sphere. If time permits, we will describe newresults on particle transport.

Houman ShokranehDept. of Aerospace EngUniversity of Southern [email protected]


MS59

Coupled Rigid-Bodies in Potential Fields

Multibody systems are modeled as two or more coupledrigid bodies that are connected and can move relative toeach other. The dynamics of such coupled rigid bodies inplanar motion in a potential field is analyzed. Dynamiccoupling between the degrees of freedom gives rise to com-plex dynamical systems that are usually not integrable. Ifthe potential field is central, then the free dynamics hasa symmetry corresponding to a cyclic variable. The freedynamics can be reduced with respect to this symmetry.Equilibria of the reduced dynamics, which correspond torelative equilibria of the full dynamics, are obtained. Thestability of these equilibria is analyzed using the energy-momentum method when it is applicable; otherwise, anexpansion of the Hamiltonian in normal form is used. Thequestion of integrability of such systems can also be an-swered in some special cases, using normal form expan-

148 DS05 Abstracts

sions.

Anthony BlochUniversity of MichiganDepartment of Mathematicsabloch@umich,edu

Harris McClamrochUniversity of MichiganDepartment of Aerospace [email protected]

Amit SanyalArizona State UniversityMechanical and Aerospace [email protected]

MS60

Cellular Automaton Modeling of Biological PatternFormation

Andreas DeutschTechnische Universitaet DresdenZentrum fuer [email protected]

MS60

A Cell-Oriented Approach to Developmental Mod-eling

The patterns of gene expression are only part of the com-plex set of processes that govern the formation of tissuestructures during embryonic development. Cells need todifferentiate and to migrate long distances through tis-sues. How do they know what to become and where togo? Cells secrete and follow gradients of diffusible chem-icals (chemotaxis) and secrete non-diffusing extracellularmatrix. In addition, variable adhesion molecules expressedon cells’ surfaces help them to form coherent structuresby differential adhesion. CompuCell is a public domainmodeling environment which implements a simple, energyminimization framework to describe these and related mor-phogenetic processes.

James A. GlazierIndiana University, Biocomplexity InstituteDepts. of Physics and Biology, School of [email protected]

Roeland MerksIndiana UniversityBiocomplexity [email protected]

MS60

Cell-Based Model for Chondrogenic Patterning inthe Chick Limb Bud

Maria KiskowskiVanderbilt UniversityDepartment of [email protected]

MS60

Modelling Aspects of Vascular Cancer

The modelling of cancer provides an enormous mathemat-ical challenge because of its inherent multi-scale nature.For example, in vascular tumours, nutrient is transportedby the vascular system, which operates on a tissue level.However, it effects processes occuring on a molecular level.Molecular and intra-cellular events in turn effect the vas-cular network and therefore the nutrient dynamics. Ourmodelling approach is to model, using partial differentialequations, processes on the tissue level and couple these tothe intercellular events (modelled by ordinary differentialequations) via cells modelled as automaton units. Thusfar,within this framework we have modelled structural adapta-tion at the vessel level and we have modelled the cell cyclein order to account for the effects of p27 during hypoxia.These preliminary results will be presented.

Philip MainiCentre for Mathematical BiologyUniversity of [email protected]

MS60

Application of Individual-Based Models Of CellMovement to Primitive Streak Formation in theChick Embryo

Timothy J. NewmanASUDepartment of [email protected]

MS60

Chick Embryo Development: Modelling the For-mation of Primitive Streak

Bakhtier VasievUniversity of DundeeDepartment of [email protected]

MS61

Nearly Inviscid Faraday Waves in Nearly Symmet-ric Containers

Parametrically driven surface gravity-capillary waves indifferent containers with weakly broken symmetry are con-sidered. In the nearly inviscid regime, the wave amplitudesinteract with an associated streaming flow, and the slowdynamics must be described by a set of coupled amplitude-streaming flow equations. This interaction can destabilizepure standing oscillations and give rise to complex time-dependent dynamics at onset. The results will be explainedand related with experiments (Simonelli and Gollub, J.Fluid Mech. 199, 471, 1989. Feng and Sethna, J. FluidMech. 199, 495, 1989).

Edgar KnoblochPhysics DepartmentUC [email protected]

Maria HigueraE. T. S. I. AeronauticosUniv. Politecnica de Madrid

DS05 Abstracts 149

[email protected]

MS61

Coupled Amplitude-Mean Flow Equations forFaraday Waves

Amplitude equations for nearly inviscid Faraday wavesmust take into account the mean flow driven in the nomi-nally inviscid bulk by Reynolds stresses generated in oscil-latory viscous boundary layers at the rigid walls and thefree surface. This flow in turn interacts with the wavesresponsible for the presence of these boundary layers. Themean flows enter the theory at the same order in pertur-bation theory as all other nonlinear terms, even in the zeroviscosity limit. The resulting description of the systemconsists of amplitude equations for the waves coupled toa Navier-Stokes-like equation for the streaming flow withboundary conditions obtained by matching the bulk flowto the boundary layer solutions, and represents a new classof pattern-forming systems. In this talk I will discuss someof the properties of this system, both in small and largeaspect ratio domains. The following speakers will describeparticular cases of this system, focusing on drift instabili-ties and relaxation oscillations.

Edgar KnoblochDepartment of PhysicsUniversity of California at [email protected]

Jose Manuel VegaUniversidad Politecnica de [email protected]

Maria HigueraE. T. S. I. AeronauticosUniv. Politecnica de [email protected]

MS61

Viscous Mean Flows in Faraday Waves: Drift In-stabilities

The weakly nonlinear dynamics of nearly inviscid Faradaywaves have been shown to be coupled to the associated vis-cous mean flow, also called streaming flow, which is pro-duced by time-averaged Reynolds stresses at the oscilla-tory boundary layers attached to the solid walls and thefree surface. This coupling has a O(1) effect in the dynam-ics beyond threshold. A good example to illustrate this isthe drift instability of spatially constant waves in annularcontainers, experimentally encountered by Douady, Fauve& Thual (1989), which have been recently shown to be dueto a reflection symmetry breaking of the mean flow.

Elena MartinUniversidad de [email protected]

Jose Manuel VegaUniversidad Politecnica de [email protected]

MS61

Localized Patterns and Mean Modes

Many different physical circumstances can generate meanmodes or mean flows. In a pattern-forming system, these

mean modes can alter the formation of patterns, leadingto chaotic or localized patterns. In one dimension, pat-terns can be destabilised by a mean mode, leading to large-amplitude localized solutions, if the coupling between themean mode and the pattern is sufficiently strong. Hexag-onal patterns, however, can be unstable for any couplingstrength, and form small-amplitude localized states.

Paul MatthewsUniversity of [email protected]

Stephen CoxUniversity of [email protected]

Michael ProctorUniversity of [email protected]

MS61

Mean Flow in (Rotating or Non-Rotating)Rayleigh-Benard Convection

Mean flows are found to be important in Rayleigh-Benardconveciton. In particular, mean flows are found in pat-tern formation near convection onset, and also in turbulentregime for large supercriticality. In the first half of this talkwe present an overview on mean flow in Rayleigh- Benardconvection. In the second half we present recent work onhow mean flow affects (or leads to) the chaotic dynamics inrotating convection. We also discuss how mean flow maymodify the defect velocity statistics.

Hermann RieckeNorthwestern [email protected]

Yuan-Nan YoungDepartment of Mathematical [email protected]

MS62

Discrete Bogdanov-Takens Bifurcation

Vassili GelfreichUniversity of Warwick, [email protected]

MS62

Bose-Einstein Condensates in the Presence Of aMagnetic Trap and Optical Lattice

Todd KapitulaUniversity of New MexicoDept of Math & [email protected]

MS62

Discrete Solitary Waves: Some Old Aspects, SomeRecent Results and Some Future Perspectives

In this short presentation, we will discuss some of the sys-tems where discrete nonlinear wave equations apply, such

150 DS05 Abstracts

as ones in atomic physics and nonlinear optics. We ’ll fo-cus on a prototypical model, namely the discrete nonlin-ear Schrodinger equation, that is relevant to these settingsand present, for each of the above two areas, an exampleof mathematical analysis/numerical computation and thecorresponding insights/results that it generated in recentexperiments. Time permitting, we’ll also examine somevery recent results on the existence and linear stability ofsuch waves and compare them to numerical findings. Fi-nally, the prototypical unit of a discrete lattice, namely adouble-well potential will also be touched upon and analyt-ical, numerical and experimental results will be presentedin that context.

Panayotis KevrekidisUMass, AmherstDept of [email protected]

MS62

Dissipationless Shock Waves in a Discrete Nonlin-ear Schroedinger Equation

It is shown that the generalized discrete nonlinearSchrodinger equation can be reduced in a small ampli-tude approximation to the KdV, mKdV, KdV(2) or thefifth-order KdV equations, depending on values of the pa-rameter. In dispersionless limit these equations lead towave breaking phenomenon for general enough initial con-ditions, and, after taking into account small dispersion ef-fects, result in formation of dissipationless shock waves.The Whitham theory of modulations of nonlinear waves isused for analytical description of such waves. The talk isbased on the works done in collaboration with M. Salerno,A. M. Kamchatnov and A. Spire, who are gratefully ac-knowledged.

Vladimir KonotopUniversidade de Lisboa,[email protected]

MS62

Existence and Stability of Embedded Solitons in aDiscrete χ(2): χ(3) Model

We report numerical findings of embedded solitons in adiscrete χ(2): χ(3) model. The model not only representsa discretization of a continuous optical problem but alsohas a straightforward physical realization of an array ofoptical waveguides. Embedded solitons are computed assymmetric homoclinic orbits to a saddle-center at the ori-gin for a four-dimensional reversible map. Two computersoftwares called AUTO and HomMap are used for the compu-tation. Moreover, the linear stability of these embeddedsolitons is determined by direct numerical integration ofthe variational ODE around them.


MS63

Oscillations, Resonances and Synchrony in the Net-works of Map-Based Model Neurons

We developed a new approach for analysis of complex large-

scale neurobiological networks based on using a system ofdifference equations to describe a neuron dynamics. Theexistence of intrinsic resonances of reliability of firing andthe effects of those resonance properties on collective be-havior in a cortical network models were studied. We showthat network interactions can enhance the frequency rangeof reliable responses and that the resonance boundaries canbe controlled by synaptic strength.

Maxim BazhenovSalk [email protected]

Nikolai RulkovUniv of California / San DiegoInst for Nonlinear [email protected]

MS63

Deriving Phase Models From Data

Phase models represent a simple and powerful way of re-ducing complex neural models to simple equations on thecircle. The key to this reduction is the so-called phase-resetting curve (PRC) which describes how the oscillatorresponds to inputs. We discuss two different methods forextracting the PRC from neural firing data. We use thePRC to create maps for instant and synaptic firing betweenneurons and analyze the resulting equations.

Bard ErmentroutUniversity of Pittsburgh,Pittsburgh, [email protected]

MS63

Patterns of Activity in Neural Tissue With Spa-tially Modulated Connectivity

In my talk I will focus on the dynamics in networks con-sisting of an excitatory and an inhibitory population ofneurons with spatially decaying connectivity. I will showthat the presence of delays give rise to a wealth of bifur-cations and to a rich phase diagram. I will show that theresults derived in that framework allow us to understandthe origin of the diversity of dynamical states observed inlarge networks of spiking neurons.

David HanselLab. de Neurophys. et de Physiol. du Systeme MoteurUniversite Rene [email protected]

MS63

Reproducible Sequence Generation in RandomNeural Ensembles

Little is known about the conditions that neural circuitshave to satisfy to generate reproducible sequences. Evi-dently, the genetic code cannot control all the details ofthe complex circuits in the brain. In addition, most of theanalysis in networks of dynamical systems deal with thedynamics within attractors. Nevertheless, sensory systemsare known to process information during transient dynam-ics. Here, We investigate the conditions on the connectivitydegree that lead to reproducible and robust sequences in aneural population of randomly coupled excitatory and in-hibitory neurons. We found that rhythmic sequences can

DS05 Abstracts 151

be generated if random networks are in the vicinity of anexcitatory-inhibitory synaptic balance. Reproducible tran-sient sequences, on the other hand, are found far from asynaptic balance. (Results to be published in Huerta, Ra-binovich PRL DEC 2004)

Ramon HuertaInstitute for Nonlinear Science, UCSD,San Diego, [email protected]

Mikhail I. RabinovichUniversity of California, San DiegoInstitute for Nonlinear [email protected]

MS63

Which Model to Use for Cortical Spiking Neurons?

We review the biological plausibility and computationalefficiency of eleven most useful and widely used modelsof spiking and bursting neurons. Our goal is to iden-tify a model that is most applicable to large-scale simu-lations of cortical neural networks. We discuss why theintegrate-and-fire neuron, being the simplest and the mostefficient spiking model, is not appropriate for simulationsand should be avoided by all means. Finally, we discusssome reasonable alternatives.

Eugene M. IzhikevichThe Neurosciences [email protected]

MS64

Integrated Diagnostic and Prognostic Health Man-agement For Mechanical and Structural SystemsWith Applications to Commercial and Defense Sys-tems

The nation’s commercial and defense manufacturers aresearching for new ways to manage product life-cycles to in-crease aftermarket profits and enable net-centric warfare.Diagnostics aim to identify faults and prognostics aim topredict the nonlinear evolution of those faults. The advan-tages and challenges of assessing the health of mechanicalsystems online are demonstrated using physics-based anddata-driven dynamic models in numerous applications in-cluding an air-handling valve, laminated composite armorspecimen and thermal protection system panel.

Douglas AdamsPurdue [email protected]

MS64

Slow-Time Damage Trajectory ReconstructionBased on Smooth Orthogonal Decomposition

Damage is viewed as evolving slowly in a hierarchical dy-namical system causing parameter drifts in a fast-timesubsystem. The fast-time measurements are used to re-construct the slow-time phase space trajectory of damage.Damage tracking feature vectors are developed based onthe phase space warping concept. Short-time evolution foreach point in the fast-time phase space is used to developfeature vectors. The hidden damage is identified by apply-ing the smooth orthogonal decomposition to these vectors.

David Chelidze

Department of Mechanical EngineeringUniversity of Rhode [email protected]

MS64

Damage Detection of Chaotically-Excited Un-manned Aerial Vehicle Wing

Interlaminar failure is a major failure process in carbon-carbon composites, which are commonly used in unmannedaerial vehicles (UAVs). To automatically assess this failuremode, we excite a UAV wing with a Lorenz driver, and ap-ply a discriminator based on a feature vector consisting ofseveral measures, including nonlinear cross-prediction er-ror, chaotic amplification of attractor distortion, continuitymeasures, as well as other time series analysis measures.

Linda J. MonizU.S. Geological [email protected]

Mary Ann F. HarrisonInstitute for Scientific Research, [email protected]

Lou PecoraNaval Research [email protected]

Steve TrickeyNaval Research LabWashington DC, [email protected]

Jon NicholsNaval Research LabWashington, DC, [email protected]

Steve Knudsen, Leon LuxemburgInstitute for Scientific Research, [email protected], [email protected]

Mark SeaverCode 5673Naval Research [email protected]

MS64

Damaging Differentiability: The Connection Be-tween Structural Damage and Loss of Synchroniza-tion

We employ chaotic interrrogation of a metal plate with in-cremental damage in order to test for changes in the struc-ture. We use two geometric tests to compare attractorsembedded from response data from an undamaged struc-ture and a damaged structure. Both embeddings take ad-vantage of a singular value decomposition of the data frommultiple sensors. The geometric methods we use are reallytests for generalized and differentiable synchronization be-tween reponses to an identical drive signal. We show thatthe test for differentiable syncrhonization shows greatersensitivity than the test for generalized synchronization, in-dicating that loss of differentiable synchronization betweenresponses is the result of damage to the structure.

Linda J. Moniz

152 DS05 Abstracts

U.S. Geological [email protected]

Louis M. PecoraNaval Research [email protected]

Jonathan NicholsNaval Research LabCode [email protected]

Steven TrickeyCoce 5673Naval Research [email protected]

Mark SeaverCode 5673Naval Research [email protected]

Daniel PecoraVirginia Commonwealth [email protected]

MS64

Detecting Damage-Induced Nonlinearities In thePresence of Ambient Variation: An Information-Theoretic Approach

Two different information-theoretics, the time-delayed mu-tual information and time-delayed transfer entropy, areused to detect damage-induced nonlinearities in struc-tures. For linear structures, both quantities admit analyt-ical treatment. An algorithm is described for estimatingboth metrics and is shown to be in agreement with theory.Damage is then introduced as a structural nonlinearity. Bycomparing the IT metrics for the structural response dataand their linear surrogates the presence of the nonlineardamage is detected.

Jonathan NicholsNaval Research LabCode [email protected]

Luke Overbey, Colin Olson, Michael ToddUniversity of California, San [email protected], [email protected], [email protected]

MS64

Structural Damage Assessment Using StochasticProbes and Pseudo-Attractor Geometry

Recent research has shown the utility of using chaotic ex-citation and state space attractor features in structuralhealth diagnostics. Attractors are reconstructed from timeseries measured simultaneously at different structural loca-tions, predictive maps are built to correlate the attractors,and an error metric is formed as the metric of interest. Thiswork considers extending this idea to stochastically-drivensystems and includes new features drawn from notions ofinterdependence and coupling.

Jonathan NicholsNaval Research LabCode 5673

[email protected]

Michael ToddUniversity of California, San [email protected]

Colin OlsonDept. of Structural EngineeirngUniversity of California, San [email protected]

Luke OverbeyDept. of Structural EngineeringUniversity of California, San [email protected]

MS65

Noise Induced Dimension Changing Bifurcations

Dramatic dynamical systems changes may occur in thepresence of noise, such as bifurcations form low dimen-sional stochastic behavior to high dimensional chaos. I willexplore this transition to illustrate some of the universalcharacteristics on a problem from continuum mechanics.

Lora BillingsMontclair State UniversityDept. of Mathematical [email protected]


MS65

Multistability, Noise and Attractor Hopping

Multistability, noise and attractor-hopping: The role ofchaotic saddles External noise applied to systems with amultitude of coexisting attractors leads to a hopping dy-namics between various metastable states. In particular,we study the role of the chaotic saddles for the accessibil-ity of different states and show how bifurcations of chaoticsaddles lead to a change in the hopping dynamics. Further-more we demonstrate an enhancement of the noise-inducedescape from attractors due to the existence of chaotic sad-dles embedded in the open neighborhood of an attractor.

Ulrike FeudelUniversity of OldenburgICBM, Theoretical Physics/Complex [email protected]

MS65

Stochastic Resonance With Frequency Sensitivity

Usually, stochastic resonance with external periodic forc-ing is not sensitive to the forcing frequency. On the otherhand, it has been well known that stochastoc resonancecan appear in autonomous excitable systems, called as co-herent resonance (CS). We find that systems of CS canshow stochastic resonance with sensitive frequency depen-dence when they are forced by periodic forcings. Moreover,the frequency-dependent stochastic resonance effect can begreatly enhanced when many excitable system are coupled

DS05 Abstracts 153

together.

Gang HuBeijing Normal University, [email protected]

MS65

Characterization of stochastic resonance

Traditional quantities used to characterize stochastic res-onance possess the common feature of low sensitivityto noise variation in the sense that they vary smoothlyabout the optimal noise level. In potential applications ofstochastic resonance such as device development, a highsensitivity to noise may be required. Here we show that,when the resonance is regarded as a manifestation of phasesynchronization, the average synchronization time betweenthe input and output signal has an extremely high sensitiv-ity in that it exhibits a CUSP behavior about the optimalnoise level. Theoretical analysis and numerical evidenceare provided to establish the cusp behavior and its gener-ality.

Kwangho Park, Ying-Cheng LaiArizona State UniversityDepartment of [email protected], [email protected]

MS65

Strange Nonchaotic Attractors in Random Dynam-ical Systems

Whether strange nonchaotic attractors (SNAs) can occurtypically in dynamical systems other than quasiperiodi-cally driven systems has long been an open question. Herewe show, based on a physical analysis and numerical evi-dence, that robust SNAs can be induced by small noise inautonomous discrete-time maps and in periodically drivencontinuous-time systems. These attractors, which are rel-evant to physical and biological applications, can thus beexpected to occur more commonly in dynamical systemsthan previously thought.

Choy Heng Lai, Xingang WangNational University of [email protected], [email protected]


MS66

Detection of Multiple Time Scale Synchronization

Using a recent procedure for analyzing nonlinear and non-stationary signals we decompose a time series in distinctoscillation modes. When applied to coupled oscillators,we found that synchronization arises in a finite number ofcommon modes. For a given coupling value, different phe-nomena as phase slips, anti-phase or perfect phase lockingcan be simultaneously observed at specific time scales. Im-plications for the study of the build-up of synchronizedstates in nonstationary and noisy systems are pointed out.

Stefano BoccalettiIstituto Nazionale di Ottica [email protected]

Mario ChavezIstituto Nazionale di Ottica AppplicataFlorence, [email protected]

Jacques MartinerieLENA-CNRS-UPR-640Hopital de la Salptriere, Paris. [email protected]

MS66

Stability of Synchronization in Networks of SpikingNeurons

In a network of neuronal oscillators with time-delayed cou-pling, we uncover a phenomenon of enhancement of neuralsynchrony by time delay: a stable synchronized state existsat low coupling strengths for significant time delays. Byformulating a master stability equation for time-delayednetworks of spike-burst Hindmarsh-Rose neurons, we showthat there is always an extended region of stable syn-chronous activity corresponding to low coupling strengths.Such synchrony could be achieved in the undelayed systemonly by much higher coupling strengths. A subset of Lya-punov exponents associated with transverse directions tothe synchronized manifold can become negative from pos-itive at different coupling strengths, indicating synchronyon different time-scales of oscillations. These results sug-gest that synchronization of spike-burst activity is a multi-time scale phenomenon and burst synchrony is easier toachieve than spike synchrony.

Viktor JirsaFlorida Atlantic UniversityPhysics [email protected]

MS66

Synchronization of Complex Brain Systems

I will present an analysis of the formation of synchronizedclusters in active dynamical networks where the activenodes are modeled by typical neuron models resp. popula-tions of neurons. The connectivity patterns are taken frommeasurements of different mammal cortex areas. Com-plex synchronization phenomena are formed and destroyedthrough (varying) interactions between the nodes. Con-ditions for optimum synchronizability are presented andinterpreted in terms of neural tasks.

Juergen KurthsUniversitat Potsdam, [email protected]

MS66

Analysis of Functional Connectivity Via HumanEEG

Functional connectivity in human neural systems is a re-search area that has attracted much interest in recent yearsdue to technological advances in equipment for collectinghuman neural data. This talks focusses on developing andanalyzing coupled cell models of human neural activity inorder to understand the mechanisms leading to connectiv-ity between neural regions and to suggest new techniquesfor detecting such interactions in real data.

John R. Terry

154 DS05 Abstracts

Department of Mathematical SciencesLoughborough [email protected]

MS66

Surrogates to Test for Synchronisation

The concept of synchronisation and especially phase syn-chronisation (PS) has been intensively studied in the re-cent years. The corresponding studies are usually basedon the computation of a measure which quantifies depen-dencies of the instantaneous phases of the time series (TS).However, even though these measures may be normalised,experimental TS yield values which are not at the bordersof the interval and hence are difficult to interpret. Thisproblem can be overcome if the coupling strength betweenthe two systems can be varied systematically and a ratherlarge change in the measure can be observed. The PS innatural systems, e.g. of heart beats of a mother with herfoetus, frequently evades such an experimental manipula-tion. In the case of mother-foetus heart beat synchronisa-tion, this problem has been tackled using ECGs obtainedfrom a group of other pregnant women (which are uncou-pled from the foetus and hence not in PS) as “naturalsurrogates” and then performing an hypothesis test. Buteven this rather innovative approach has some drawbacks.The natural variability and also the frequency of the heartbeats of the surrogate mothers is usually different from theones of the real mother. Furthermore, the data acquisitioncan be expensive and at least in some states of the preg-nancy problematic. In these cases it is highly convenientto generate the surrogates by a mathematical algorithm.We present an approach for the generation of surrogates,which is based on the recurrences of a system. These sur-rogates mimic the dynamical behaviour of the system andovercome the above mentioned problems.

Marco ThielInstitut fur PhysikPostdam University, [email protected]

Maria Carmen RomanoAm Neuen Palais,1014469 Potsdam, [email protected]

Jurgen KurthsAm Neuen Palais,[email protected]

MS66

Simulation of EEG Properties with Random Net-works of Leaky Integrator Neurons

The human electroencephalogram (EEG) is globally char-acterized by a 1/f power spectum superimposed by certainpeaks where the alpha peak around 8 - 14 Hz is the mostprominent one. The spectral power in the alpha band de-pends on several variables: increased arousal leads to adecrease of the alpha power (“alpha block”) and the powerincreases during children development accompanied by ashift from lower frequencies. I will show how some of thesecharacterisics can be modeled by evolving random networksof leaky integrator units where a critical connectivity mustbe reached for the onset of oscillations in the network.

Peter beim GrabenUniversity of Potsdam

[email protected]

MS67

Fronts in Media with Nonlocal Interaction

We consider various lattice dynamical systems with longrange interaction and related integro-differential evolutionequations. Typically, these arise in the modeling of phasetransitions for a binary material and from activity in fam-ilies of neurons. Nonlocal analogs of the wave equation,Allen-Cahn and Cahn-Hilliard equations, and the phase-field system will be discussed.

Peter BatesMichigan State UniversityDepartment of [email protected]

MS67

Traveling Fronts in Scalar Reaction-DiffusionEquations

We prove existence of a family of traveling wave solutionsfor a large class of scalar reaction-diffusion equations withdegenerate, nonlinear diffusion coefficients and monostablenonlinear reaction terms. We also show that, as in thelinear diffusion case, the slowest traveling wave solution inthe family yields the asymptotic rate of the propagation ofdisturbances from the unstable rest state in these systems.In addition, we give conditions on the reaction term anddiffusion coefficient ensuring existence of interfaces.

Georgi MedvedevDepartment of MathematicsDrexel [email protected]

MS67

Large Entire Solutions of a Reaction-DiffusionEquation with Bistable Nonlinearity

We consider a reaction-diffusion equation of one-space di-mension with bistable nonlinearity and assume that it hastraveling front waves with constant speed. Then one canobserve the annihilation and diverging of two fronts for ap-propriate initial data. We show the existence of entire solu-tions realizing these phenomena, where the entire solutionis meant by a bounded solution defined for all (x, t) ∈ R2.Namely we prove that there exist two kinds of entire solu-tions such that the two fronts coming from the both sidesof x-axis annihilate while the other one emanating from anunstable equilibrium converges to the diverging fronts.

Yoshihisa MoritaRyukoku UniversityDepartment of Appl. Math. and [email protected]

MS67

A Variational Approach to Traveling Waves in Gra-dient Systems

We study a class of systems of reaction-diffusion equa-tions in infinite cylinders. These systems of equations arisewithin the context of Ginzburg-Landau theories and de-scribe the kinetics of phase transformation in second-orderor weakly first-order phase transitions with non-conservedorder parameter. We use a novel variational characteriza-

DS05 Abstracts 155

tion to study existence of traveling wave solutions undervery general assumptions on the nonlinearities. These so-lutions are a special class of the traveling wave solutionswhich are characterized by a fast exponential decay in thedirection of propagation. Our main result is a simple ver-ifiable criterion for existence of these traveling waves. Wealso prove boundedness, regularity, and some other prop-erties of the obtained solutions, as well as several sufficientconditions for existence or non-existence of such travelingwaves, and give rigorous upper and lower bounds for theirspeed. In addition, we prove that the speed of the ob-tained solutions gives a sharp upper bound for the propa-gation speed of a class of disturbances which are initiallysufficiently localized. We give a sample application of ourresults using a computer-assisted approach.

Cyrill B. MuratovNew Jersey Institute of TechnologyDepartment of Mathematical [email protected]

MS68

Projective Integration and Multiscale Problems

We discuss ways in which the macroscopic behavior can bederived from a microscopic model. The microscopic modelis used to estimate the forward derivative of macroscopicvariables for use in any process that could use a directevaluation of derivatives. We determine long-term behav-ior, steady states, estimates of the slow manifold, and even,in some cases, integrate backwards in spite of instability ofthe underlying problem.

C. William GearPrinceton [email protected]

MS68

Some Computational Examples of Equation-FreeDynamic Renormalization

I will discuss a number of modeling examples whose com-putation is facilitated in an equation-free multiscale frame-work. I will also present some examples of equation-free dynamic renormalization computations; these rangefrom simple glassy dynamics in compaction models to self-similar interface roughness evolution dynamics. Differentexamples involve collaborations with different groups, whowill be mentioned in the presentation.


MS68

Continuum-Microscopic Simulation of Chemotaxis

Continuum models of chemotactic behavior of bacteriahave typically used empirical closures to furnish trans-port coefficients. A computational approach is presentedin which microscopic simulation is carried out at the be-ginning of a continuum time step to determine local trans-port coefficient values. Adaptive mesh refinement on thecontinuum level is used to identify large gradient regions.Differing spatial resolutions on the continuum scale lead todifferent coarse grained transport coefficients which mustreconciled when carrying out fine-to-coarse grid updates.A comparison is made among different microscopic mod-els including: cellular automata, lattice Boltzmann and a

canonical dissipative system using viscous fluid singulari-ties.

Sorin M. MitranDept. of Mathematics, Applied Math Prog.University of North [email protected]

MS68

An Alternative Projective Integration Scheme

Gear et al. proposed the class of projective integrationmethods for problems with gaps in their eigenvalue spec-trum. These methods combine consecutive results of anexplicit time stepper, to obtain an extrapolated solution.We develop and analyse a variant of this method, by ex-tending the results of Sommeijer (Comput. Math. Applic.,19(6):37-49, 1990). A stability analysis of the resultingmultistep method reveals that the method can also be ap-plied to problems with no gap in their eigenvalue spectrum.

Dirk RooseK.U.LeuvenDept. of Computer [email protected]

Kurt LustUniversity of GroningenInstitute of Mathematics and Computing [email protected]

Christophe VandekerckhoveDept of Computer ScienceKatholieke Universiteit [email protected]

MS69

Dimension Reduction Techniques for Rigorous Nu-merics

Most interesting dynamical systems require a number ofdimension reductions before they can be studied numeri-cally. In order to produce rigorous results, these reductionsmust be performed in such a way that information lost maybe recovered, at least in a coarse way, by mathematicalanalysis. In this talk, we will discuss some techniques fordimension reduction and some of the mathematical toolswhich may be used to compute rigorous information aboutthe full system.

Sarah DayDepartment of MathematicsCornell [email protected]

MS69

Rigorous Numerics for the Two Dimensional Cahn-Hilliard Equation

Computing explicit solutions for partial differential equa-tions is usually a difficult task. Therefore one oftenuses the computer to calculate approximations, but inmany cases it is not clear whether this numerical datareally approximates a solution of the partial differentialequation. We present a method, based on Conley in-dex theory, that insures the existence of an equilibriumof a time dependent equation in a computed neighbor-hood. This technique was introduced by K. Mischaikow

156 DS05 Abstracts

and P. Zgliczynski for steady states of the one dimensionalKuramoto-Shivashinsky equation for fixed parameter val-ues. We improved this method in order to compute pathsof equilibria for a two dimensional problem, namely theCahn-Hilliard equation on the unit square. The Cahn-Hilliard equation, a parabolic equation of fourth order, wasintroduced as a model for the process of phase separationof a binary alloy at a fixed temperature. We summarizesome analytical results on its set of equilibria and thenpresent the results we obtained using the rigorous method.This includes secondary bifurcations from the known mainbranches of equilibria.

Stanislaus Maier-PaapeRWTH AachenInstitut fur [email protected]

Ulrich MillerTheoretical and Applied MechanicsCornell [email protected]

MS69

Computing the Dynamics of Infinite DimensionalMaps

I will discuss computational methods that can be used torigorously study the dynamics of infinite dimensional maps.The Conley index provides the theoretical underpinning forthese techniques. I will focus on the mathematical aspectsof the theory that allow one to minimize the computationalcosts.


MS69

Chain Recurrence From a Combinatorial Point ofView

In this talk we present a new definition of the chain recur-rent set of a continuous map using finite spatial discretiza-tions. This approach allows for an algorithmic construc-tion of index filtrations for Morse decompositions which ap-proximate the chain recurrent set arbitrarily closely as wellas discrete approximations of Conley’s Lyapunov function.This is a natural framework in which to develop computa-tional techniques for the analysis of qualitative dynamics.

William D. KaliesFlorida Atlantic UniversityDepartment of Mathematical [email protected]


Robert VandervorstDepartment of MathematicsVrije Universiteit, [email protected]

MS70

A Variational Problem on Stiefel Manifolds

We consider here a general class of continuous timequadratic cost, optimal control problems on Stiefel man-ifolds, which in the extreme dimensions yield the classicalrigid body equations and the geodesic flow on the ellipsoid.This is related to earlier work of Moser and Veselov in thediscrete setting. We have already shown that this optimalcontrol setting gives a new symmetric representation of therigid body flow and in this paper we extend this represen-tation to the geodesic flow on the ellipsoid and the moregeneral Stiefel manifold case.

Peter E. CrouchArizona State [email protected]

Anthony M. BlochUniversity of MichiganDepartment of [email protected]

Amit SanyalArizona State UniversityMechanical and Aerospace [email protected]

MS70

Algorithmic Mechanics on Lie Groups

Many constrained optimization problems are formulatednaturally as optimization problems on classical Lie groupsand homogeneous spaces. Considerations of faster conver-gence and better dynamical behavior lead us to embeddingcertain classical algorithms such as gradient flow in secondorder equations of mechanics. Specifically we consider me-chanical systems on Lie groups wherein the objective func-tion does double duty in determining stiffness and damp-ing. In this talk, we investigate the basic equations forcertain appealing objective functions and the asymptoticbehavior of the associated mechanical systems.

Uwe R. HelmkeUniversitat WuerzburgDepartment of [email protected]

P.S. KrishnaprasadInstitute for System ResearchUniversity of Maryland at College [email protected]

MS70

Computational Geometric Mechanics and its Ap-plications to Geometric Control Theory

The geometric approach to mechanics serves as the theo-retical underpinning of innovative control methodologies ingeometric control. These include, for example, controllingthe attitude of satellites using changes in its shape, as op-posed to chemical propulsion. We will introduce some ofthe discrete differential geometric machinery necessary toimplement control algorithms while respecting and preserv-ing the geometry of the problem. These include discreteanalogues of Lagrangian mechanics, exterior calculus, andconnections on principal bundles.

Melvin Leok

DS05 Abstracts 157

University of Michigan, Ann ArborDepartment of [email protected]

MS70

Nonholonomic Integrators on Lie Groups

Variational integrators proved to be very effective in long-term numerical simulations of holonomic mechanical sys-tems. Recently, such integrators were adapted to work inthe nonholonomic setting. This talk will give an overviewof the theory. Conservation of momentum and energy bythe numerical algorithm will be discussed. This is jointwork with Yuri Fedorov.

Dmitry V. ZenkovNorth Carolina State UniversityDepartment of [email protected]

MS71

Modeling Healthy and Pathological Neovascular-ization of the Retina

Maturation of the human fetal retina is regulated by thedelivery of oxygen to differentiating retinal cells; the devel-opment of vasculature to deliver this oxygen is regulatedby the production of VEGF by these cells in response tohypoxia. We present a model for retinal growth compris-ing coupled PDEs describing the interdependence of reti-nal differentiation and capillary distribution. Our goal isto distinguish the physiological path to retinopathy of pre-maturity from normal retinal development.

Chris GraesserDepartment of Mechanical & Industrial EngineeringUniversity of Illinois at [email protected]

Marialuisa RuizEntelos, [email protected]

Charles SimmonsDavid Geffen School of [email protected]

Scott KellyDepartment of Mechanical & Industrial EngineeringUniversity of Illinois at [email protected]

MS71

Mechanical Forces During Vasculogenesis and An-giogenesis

During blood vessel formation, endothelial cells follow me-chanical as well as chemical cues in their environment. Thetheory that describes these mechanochemical interactionsassumes that the extracellular matrix (ECM) is a viscoelas-tic material which deforms under cellular traction, and thatcell movement depends on chemoattractant gradients andECM strain. Numerical simulations predict cell tractioncan reorganize the ECM into a network. I discuss the po-tential role of chemical vs. mechanical forces during blood

vessel formation.

Daphne ManoussakiVanderbilt [email protected]

MS71

A 2-Dimensional Model of Growth Factor InducedAngiogenesis

We propose a nonlinear coupled system of partial differ-ential, ordinary differential and algebraic equations, forgrowth factor-induced angiogenesis. Properties of the ex-tracellular matrix are incorporated within the conductivity.An indicator function tracks the capillary network. A set ofconditions captures the network characteristics. Numericalresults of the capillary network growth dependence on theproperties of the growth factor (its diffusivity, decay, andconsumption rate) and the extracellular matrix (anisotropyand heterogeneity) will be discussed.

Charles PatrickUniversity of TexasM. D. Anderson Cancer [email protected]

Shuyu SunUniversity of Texas - [email protected]

Mary WheelerCenter for Subsurface ModelingThe University of Texas at Austin, [email protected]

Mandri ObeyesekereUniversity of Texas M. D. Anderson Cancer [email protected]

MS71

Numerical Simulation of Capillary Formation Dur-ing the Onset of Tumor Angiogenesis

A model for tumor angiogenesis consisting of a coupledsystem of ordinary and partial differential equations is dis-cussed. The modeling of chemotaxis of endothelial cells tobiochemicals is a key feature in the model, which resultsin convection dominated diffusion equations. A numericalmethod that efficiently treats problems of this type is pre-sented. It is based on the use of characteristics, is massconserving, and provides an effective means of evaluatingmodeling scenarios.

Marit Nilsen-HamiltonDepartment of Biochemistry, Biophysics and MolecularBiologyIowa State [email protected]

Howard Levine, Michael SmileyIowa State [email protected], [email protected]

MS72

Mathematical Model Studies of LagrangianStochastic Methods

Lagrangian Stochastic Models (LSMs) are being actively

158 DS05 Abstracts

developed as computationally tractable schemes to pre-dict and analyze oceanic transport of immersed substances,without full resolution of the oceanic turbulence. In orderto understand better how LSMs can, on a fundamentalphysical level, model certain transport features, such assubdiffusion and superdiffusion, we analyze them in thecontext of relatively simple mathematical model flows in-cluding a mean flow, low-frequency variability, and a tur-bulent component.

Emilio CastronovoRensselaer Polytechnic InstituteDept. Mathematical [email protected]


MS72

Application of Auto-Regressive models to La-grangian Prediction and Stochastic BC

Auto-Regressive (AR) models are widely used in spectralanalysis, but their use as stochastic models for geophysi-cal variables is still under development. Geophysical ap-plications of AR models include the random flight modelfor Lagrangian motion and applied Lagrangian prediction;the reduced-order information filter, based on an extendedsecond-order AR model in space, for the assimilation ofaltimetric data into ocean circulation models; coupled ARmodels for modeling Lagrangian covariance functions; andthe use of spatio-temporal AR models for parameterizingstochastic boundary conditions in coastal flows. Our stud-ies have shown that AR models are parsimonious modelsfor geophysical variables and they should be used in a largeclass of parameterization and prediction problems.

Annalisa GriffaUniversity of [email protected]

Arthur MarianoRSMAS/MPOUniversity of [email protected]

Tamay OzgokmenUniversity of Miami/[email protected]

Toshio Mike ChinJPL, California Institute of [email protected]

MS72

Parameterization of Turbulent Transport in Ma-rine Ecosystems

Statistical properties of Lagrangian trajectories in turbu-lent flows can be reasonably reproduced by using stochas-tic models of different complexity. Here, we focus on theextension of stochastic parameterization for transport inthe case the advected particles contain reactive compo-nents. We use a simple model of the marine ecosystem dy-namics (nutrient, phytoplankton, zooplankton) coupled todifferent advection models (quasi-geostrophic turbulence,

Langevin models, diffusion) to explore the effects of differ-ent transport parameterization types on primary produc-tivity and residual nutrient concentration.

Claudia PasqueroGeological and Planetary Sciences DivisionCalifornia Institute of [email protected]

MS72

Closure theory for the effective diffusivity tensor inforced beta-plane turbulence

Theoretical and numerical results for the stirring of a tracerboth along and across turbulent jets is presented. Such asystem can be taken as a model for the stirring of true trac-ers along jets in either the atmosphere and ocean, or, lessobviously, for the stirring of baroclinic potential vorticityby non-zonal flow in the ocean. The model flow consideredis two-dimensional turbulence with a mean vorticity gra-dient (β) and forced randomly and isotropically at small-scales. The tracer transported by the flow is forced by amean tracer gradient that is arbitrarily oriented with re-spect to the mean vorticity gradient. Such a tracer canbe decomposed into two independent tracers, one forcedby a gradient that is parallel to the vorticity gradient (andso is stirred across jets), and another that is forced by amean gradient that is perpendicular to the mean vorticitygradient (and so is stirred along jets). The effective dif-fusivity tensor for the full tracer can be computed fromthe eddy correlations of the two decomposed tracers. Theacross-jet diffusion is well-described by mixing length the-ory, while the diffusivity of the tracer stirred along the jetsis the result of shear dispersion. At only moderate levelsof anisotropy in the flow, the along-jet diffusivity is twoorders of magnitude larger than the across-jet diffusivity.The skew flux of the tracer is, notably, of the same magni-tude as the mean flow.

Shafer SmithCenter for Atmosphere Ocean ScienceCourant [email protected]

MS73

Decay of Chaotically Advected Passive Scalars inthe Zero Diffusivity Limit

There has been much recent interest in the time asymp-totic decay of the variance of a passive scalar advected bya chaotic flow. In particular, there is a debate as to the rel-ative importance of long wavelength and short wavelengthprocesses in determining zero diffusivity limit of the decayrate. Here we investigate the validity and regimes of appli-cability of these two types of mechanisms. We argue thatunder typical conditions the short wavelength mechanismprovides an upper bound on the decay rate in the zero dif-fusivity limit. We also demonstrate the applicability of theshort wavelength mechanism for a particular numerical ex-ample by showing that the decay rate is insensitive to theoperation of filtering out the longest wavelength perturba-tions [1]. The decay of variance in the case in which thescale of the flow field is much less than that of the scalar isgoverned by long wavelength processes. In this case there isa relation between the decay rate and the exponent in thepower law for the wavenumber dependence of the scalar’sspectrum [1,2]. Finally, the long-time decay rate of thescalar variance is sensitive to the chaotic properties of the

DS05 Abstracts 159

flow. Examples of this will be discussed.

Yue-Kin TsangIREAPUniversity of [email protected]

Edward OttUniversity of MarylandInst. for Plasma [email protected]

Thomas AntonsenInstitute for Research in Electronics and Applied PhysicsUniversity of [email protected]

MS73

Effective Dimensions and Chemical Activity inClosed Flows

We show how reactions (in particular, autocatalytic ones)spread in a closed container due to diffusive chaotic mixing.Contrary to open flows, in closed flows there are no fractalobjects forming the skeleton of activity. Nevertheless, weare characterising the process via an effective dimensionwhich is converging asymptotically towards the dimensionof the flow.

Tamas TelInstitute for Theoretical PhysicsEtovos University, [email protected]

Gyorgy KarolyiDept. of Structural MechanicsBudapest University of Technology and [email protected]

MS73

Diffusive Interfaces and Boundary Layers at HighPeclet Numbers

I will describe the behavior of solutions of a steadyadvection-diffusion problem on a bounded two-dimensionaldomain with prescribed Dirichlet data when Pe, the Pecletnumber is very large. The characteristic property of ad-vection by cellular flows is that the fluid motion is sepa-rated into flow cells by diffusive interfaces. At high Pecletnumbers advection dominates diffusion and the solutionstend to constant in each flow cell. Boundary layers of or-der Pe−1/2 arise near diffusive interfaces. In this talk Iwill discuss this boundary layer structure by means of anasymptotic model on diffusive interfaces.

Alexei NovikovPenn State [email protected]

MS73

Chaotic Advection and Persistent Patterns in Sta-tistical Mechanics

A fluid undergoing chaotic advection corresponds directlyto Liouville distribution dynamics. The persistent patternsstudied in fluids have interesting interpretation and rolesin statistical mechanics. These extremize the second rate

of change of the (Renyi) entropy of the distribution, pro-ducing entropy according to a generalized Lyapunov ex-ponent. Their properties govern the quantum to classicaltransition in decoherent quantum chaos. We discuss othersuch fundamental questions that can be explored throughexperiments in chaotic advection.

Arjendu K. PattanayakDepartment of Physics and AstronomyCarleton [email protected]

MS74

Asymptotic Properties of Equations of BiologicalExcitability

We consider various parametric embeddings of models ofbiological electrical excitability, from Hodgkin-Huxley tomodern cardiac models. Tikhonov-Pontryagin style smallparameters, with slow manifold having van der Pol’s fold,or Zeeman’s cusp catastrophe, inevitably lead to misrep-resentation of important qualitative features of the fullsystems. We introduce alternative parameterization, notamenable to Tikhonov theorem. It is based on actual prop-erties of full models and describes phenomena that evadetraditional asymptotic description.

Vadim N. BiktashevDept of Mathematical SciencesUniversity of [email protected]

Irina V. Biktasheva, Radostin D. Simitev, RebeccaSuckleyUniversity of [email protected], [email protected],rebecca [email protected]

MS74

Milti-Time-Scale Dynamical Systems and TheirSynchronization

We consider dynamical systems with two time scales whereboth fast and slow dynamics can be nontrivial (in particu-lar, chaotic). We show how a reduced slow system can bederived. When two such multi-scale systems are coupled,an effect of partial synchronization can be observed, wherethe slow variables are synchronized while the fast ones areindependent. As an example bursting dynamics of neuronsis considered.

Arkady PikovskyDepartment of PhysicsUniversity of Potsdam, [email protected]

Polina LandaMoscow State [email protected]

Michael RosenblumPotsdam UniversityDepartment of [email protected]

MS74

Complex Dynamics of Two Time Scale Neuron

160 DS05 Abstracts

Models

The talk focuses on homoclinic bifurcations of periodic or-bits and tori in singularly perturbed systems following theHodgkin-Huxley formalism. Of special interests are thebifurcational scenarios that underlie plausible biophysicaltransitions between tonic spiking and bursting regimes ofa neuron, as well as, explain its bi-stability.

Andrey ShilnikovGeorgia State UniversityDept Mathematics [email protected]

Gennady S. CymbalyukGeorgia State UniversityDepartment of Physics and [email protected]

MS74

Fast and Slow Dynamics in Gene Regulation

Recent experiments show high variability of gene expres-sion throughout the cell lifecycle which operates on vastlydifferent time scales. Some of these scales are imposedby extrinsic factors and higher level control mechanisms(e.g. circadian clock), while others are intrinsic to the geneexpression itself. Usually binding/unbinding and dimer-ization of enzymes and other molecules are fast, and thetranscription and translation steps are typically slow. Inthis talk we deduce simplified deterministic and stochas-tic equations for the slow transcription/translation dynam-ics. These equations differ significantly from the usual rateequations a priori assuming local equilibrium of fast re-actions. Using these equations, we discuss transient andnon-Markovian effects in gene regulation in simple geneticcircuits.

Dmitri VolfsonUniversity of California, San [email protected]

Jeff HastyJacobs School of EngineeringUC San [email protected]

Dmitri Bratsun, Lev S. TsimringUniversity of California, San [email protected], [email protected]

MS75

Two Dimensional SPH Simulation of Floating andSinking Objects

SPH is a Lagrangian method invented in the seventies andit has been extended to incompresible flows in the eighties.Since, results in good accordance with experiments havebeen obtained in some cases like water collapse or confinedproblems like sloshing in tanks. Nevertheless, SPH leads toinaccuracies mainly coming from the boundary treatmentand the viscosity formulation. These points are discussedhere dealing with solid objects moving in water. Differentpossible formulations are compared.

Louis DelormeEscuela Tecnica Superior de Ingenieros NavalesUniversidad Politecnica de [email protected]

MS75

Short Time Lyapunov Exponents to AnticipateVessel Instabilities

The chaotic behavior of finite-time phenomena such as ves-sel capsize, is quantifiable using a short, or finite-time Lya-punov exponent. This talk presents an investigation of fi-nite time Lyapunov exponent time series for vessel capsizein regular beam seas calculated from experimental datausing a combined numerical-experimental approach. Ad-ditionally, the details of the Jacobian evaluation in thecombined method and comparison to finite-time Lyapunovexponents based upon numerical simulation are discussed.

Armin TroeschNaval Architecture and Marine EngineeringUniversity of [email protected]

Leigh MccueDepartment of Aerospace and Ocean EngineeringVirginia [email protected]

MS75

Capsize Assessment Using Modern Methods

Modern methods are analysis and modeling approxima-tions applied to complex real systems. The modeling re-ductions necessary to produce tractable equations are fre-quently called into question for physical relevancy. Thistalk reviews several such examples and suggests ways inwhich critical insight can be gained by viewing model cap-size experiments as extensions of more idealized nonlinearsystems.

Armin TroeschNaval Architecture and Marine EngineeringUniversity of [email protected]

MS75

Theoretical and Experimental Studies of PassiveNnonlinear Targeted Energy Transfer in Systemsof Coupled Oscillators

We study the dynamics of passive energy transfer fromdamped linear oscillators to essentially nonlinear end at-tachments. This transfer is caused by fundamental or sub-harmonic resonance captures, and in some cases is initi-ated by nonlinear beat phenomena. The end attachmentsare capable of passively absorbing broadband energy bothat high and low frequencies, acting, in essence, as passivebroadband boundary controllers. Complex transitions dueto bifurcations of periodic orbits, and experimental resultsare discussed.

Alexander VakakisDepartment of Applied Mathematical and PhysicalSciencesNational Technical University of [email protected]

MS76

An Overview of Normal Form and UnfoldingStyles.

x = Ax + · · · is in normal form if the higher order terms

DS05 Abstracts 161

belong to a complement of the space of terms removablethrough coordinate changes in each degree. A normal formstyle is a choice of the complementary space. There are fourstyles in common use: Poincare-Dulac, inner product, sim-plified, and sl(2). We will review the definitions, advan-tages and disadvantages, and state of current knowledgeabout these styles, extend the notion of style from nor-mal forms to (asymptotic) unfoldings, and indicate how tocompute these unfoldings in any style.

James A. MurdockIowa State UniversityDepartment of [email protected]

MS76

A Normal Form Of Thin Fluid Film Equations Pro-vides Initial Conditions

We construct a normal form of the Navier–Stokes equationsfor the flow of a thin layer of fluid upon a solid substrate.This illuminates the fluid dynamics by decoupling the in-teresting long-term ‘lubrication’ flow from the rapid viscousdecay of shear modes. The normal form clearly shows thecentre manifold of the lubrication model and demonstratesthat the initial condition for the fluid thickness of the lubri-cation model is not the initial physical fluid thickness, butinstead is modified by the initial lateral shear flow. Withthese initial conditions, better forecasts will be made usingthe lubrication model.

Anthony RobertsUniversity of Sothern QueenslandToowoomba 4352, [email protected]

MS76

Towards Unique Normal Form Computation WithQuadratic Convergence

The computation of the transformation into normal formof a vectorfield at equilibrium with respect to its linearpart can be done with quadratic convergence, but this isno longer true for the computation to higher order (hyper)normal form. In this talk I’ll describe the obstructionsto this in terms of spectral sequences and show how onecan try and find optimal convergence somewhere betweenlinear and quadratic.

Jan SandersFree University of AmsterdamTne [email protected]

MS76

Normal Forms and Averaging For Partial Differen-tial Equations

The techniques of averaging normal forms are well estab-lished for ordinary differential equations, but far from com-plete in the case of partial differential equations. This talkis a report on the current status of rigorous averaging andnormal form theory for partial differential equations. Forbounded domains and operators allowing for a semigroupformulation we can formulate normal forms and approxi-mation theorems. In the case of wave operators for un-bounded domains we can apply averaging over the char-acteristics but the corresponding approximation theorems

are weaker and are probably in need of new methods.

Ferdinand VerhulstUniversity of UtrechtMathematics [email protected]

MS77

Euclidean Shift-Twist Symmetry in PopulationModels of Molecular and Cellular Alignment

We consider the symmetry properties of a general classof nonlocal population models describing the aggregationand alignment of oriented objects in two dimensions. Suchobjects could be at the level of molecules, cells or wholeorganisms. We show that the underlying interaction kernelis invariant under the shift-twist action of the Euclideangroup acting on the space R2×S1. We then use equivariantbifurcation theory to identify the types of spatio-angularpatterns that are expected to occur, and compare thesewith experiments.

Paul BressloffUniversity of [email protected]

MS77

Deriving Information About Architecture FromActivity Patterns In Coupled Cell Systems

We take an inverse approach to the study of coupled net-work activity, asking how the existence of a specified ac-tivity pattern constrains the possible network coupling ar-chitectures. For patterns featuring multiple synchronizedclusters, we derive a linear relation, the solutions of whichare precisely those architectures that are robustly compat-ible with a given pattern. The analysis of this relationshows how the inclusion of a second “hidden” cell popula-tion can broaden the range of architectures that supportparticular activity patterns in an “observed” population.

Jonathan E. RubinUniversity of PittsburghDepartment of [email protected]

Kresimir JosicUniversity of HoustonDepartment of [email protected]

MS77

Unstable Attractors?Prevalence in Networks of Pulse-Coupled Oscilla-tors

Attractors are central to studies in many fields of science,because they determine the long-term behavior of most dis-sipative dynamical systems. Traditionally, attractors areviewed as being asymptotically stable invariant sets whichhave a neighborhood that absorbs all sufficiently close ini-tial conditions. In 1985, John Milnor introduced a con-cept of attractor that neither presumed nor implied sta-bility. Here we show that unstable attractors exist andeven occur robustly in models of biological synchronizationphenomena, namely in networks of oscillators with delayedpulse-coupling (Timme et al., Phys. Rev. Lett. 89:154105,2002). From random initial conditions, groups of synchro-

162 DS05 Abstracts

nized oscillators (clusters) are formed that send pulses al-ternately, resulting in a periodic dynamics of the network.The full measure of arbitrarily weak perturbations causesa splitting-up of certain clusters such that trajectories de-part from the attractor. This is explained by the surpris-ing geometrical fact that these unstable attractors are sur-rounded by basins of attraction of other attractors whereasthe full measure of their own basin is located remote fromthe attractor. Unstable attractors do not only robustly ex-ist in these systems but moreover dominate the dynamicsfor large networks and a wide range of parameters.

Marc TimmeMax Planck Institut for Dynamics and Self-OrganizationGoettingen, [email protected]

MS77

Robust Patterns in Coupled Cell Systems

A coupled cell system is a network of dynamical systems, or“cells”, coupled together. For such systems, we investigatepatterns whose spatio-temporal symmetries are preservedunder small changes in the equations describing the cells.We conjecture that these patters are the consequence of anunderlying symmetry of the system, and prove this in somecases.


Ian StewartUniversity of WarwickMathematics [email protected]

Andrew TorokUniversity of HoustonDepartment of [email protected]

MS78

Holes in Oscillatory Wave Propagation

Holes are almost planar interfaces for which the angles ofthe interface at each point, relative to a fixed planar inter-face, tend to zero at infinity. In isotropic reaction-diffusionsystems holes bifurcate from stable planar pulsating fronts.We show that their dynamics can be formally describedby a quadratic system of viscous conservation laws for an-gle and frequency. In this setup holes are homoclinic or-bits which decay algebraically at infinity. We use blow-upmethods to construct an Evans function and study the sta-bility problem.


Mariana HaragusLaboratoire de Mathematiques de BesanconUniversite de Franche-Comte, [email protected]

MS78

Panel Discussion

The minisymposium presented a variety of examples, wheregeometric blowup both made a rigorous stability analysispossible and provided a ’typical’ geometric picture of theexistence and stability problem. The discussion will fo-cus on three aspects: - comparison of the geometric viewto matched asymptotic expansions - numerical implemen-tation of the blowup algorithm - open problems and newdirections


Mariana HaragusLaboratoire de Mathematiques de BesanconUniversite de Franche-Comte, [email protected]

MS78

Evans Function and Blow-Up for Degenerate ShockWaves

We consider the Evans function approach to the stability ofviscous shock waves in the case of characteristic shocks, i.e.shocks with shock speed equal to a characteristic speed atone of the end states. In comparison to the well-understoodnon-characteristic case two complications arise: first, theslow (merely algebraic) decay of the traveling wave at thecharacteristic end state; second, the fact that the essentialspectrum of the corresponding linearized equation has abranch point at the origin. We show how an Evans functioncan still be meaningfully defined in that case by means ofthe blow-up technique.

Nikola PopovicBoston UniversityCenter for BioDynamics and Department of [email protected]

MS78

Zero Versus Nonzero Contact Angles in Thin FilmProblems

In the lubrication approximation, thin films are describedby a degenerate parabolic equation for the height ht =(hnhxxx)x. For 0 < n < 3, self-similar solutions exist withdecreasing maximal height and spreading, compact sup-port. At the boundary of the film, the surface touchesdown with either zero or arbitrary finite contact angle.We investigate the linearized stability of these solutionsexploiting the scaling invariance and a sequence of trans-formations. The geometric picture in the new coordinatesexhibits a striking analogy to nonlinear front propagationin reaction-diffusion systems and the conjecture for the se-lection of the front with steepest decent.


Don AronsonUniversity of [email protected]

DS05 Abstracts 163

MS78

Stability of Viscous Shock Waves

Evans function methods are a powerful tool in the analysisof the stability of viscous shock waves. However, at variousstages in the analysis problems associated with the lack ofhyperbolicity in the underlying ODEs must be overcome.In this talk we show how rescaling and blow-up methodscan be used to overcome these difficulties. This is jointwork with H. Freistuhler (Leipzig).

Peter SzmolyanInstitute for Analysis and Scientific ComputingVienna University of [email protected]

MS78

The Generalized Korteweg-de Vries Equation: Sta-bility of Solitary Waves in the Singular Speed Limit

Pego and Weinstein studied the gKdV equation, calculatedthe Evans function and gave conditions for spectral stabil-ity. This theory cannot be applied in the singular limitc = 0 of the wave speed (a standing wave) because theEvans function is not defined anymore. The main reason isthat the absolute spectrum touches the essential spectrumand the profile loses exponential dichotomy properties, i.e.the profile of the standing wave is just algebraically decay-ing. We are able to analyse the stability properties of thestanding wave by applying the blow-up technique whichallows us to extend the Evans function into this singularlimit.

Bjorn SandstedeUniversity of [email protected]



MS79

Navier Stokes: Invariant Measures and Cascades

Randomly forced Navier-Stokes system in 3D will be dis-cussed. I will give a new existence-uniqueness theorem forstationary solutions under some smallness conditions onthe random forcing. The stationary solution will be con-structed and studied with the help of a beautiful stochasticcascade construction due to Le Jan and Sznitman.

Yuri BakhtinDuke [email protected]

MS79

Ergodicity and Stochastic PDEs

Martin HairerWarwick University, United [email protected]

MS79

Long-time Asymptotics of a Multiscale Model forPolymeric Fluid Flows

We investigate the long-time behaviour of some micro-macro models for polymeric fluids (Hookean model andFENE model), in various settings (shear flow, generalbounded domain with homogeneous Dirichlet boundaryconditions on the velocity, general bounded domain withnon-homogeneous Dirichlet boundary conditions on the ve-locity). We use both probabilistic approaches (couplingmethods) and analytic approaches (entropy methods).

Tony LelievreUniversity of Montreal, [email protected]

MS79

Surprising Dissipation in Models of Inviscid Limits

Jonathan C. MattinglyDepartment of MathematicsDuke [email protected]

MS79

Nonequilibrium Stationary States in Classical Sta-tistical Mechanics

Luc Rey BelletDepartmant of MathematicsUniversity of [email protected]

MS79

Invariant Measure of Stochastic PDEs and Condi-tioned Diffusions

Eric Vanden-EijndenCourant [email protected]

MS80

Noise-Induced Superpersistent Chaotic Transients

Superpersistent chaotic transients are characterized by anexponential-like scaling law for their lifetimes where the ex-ponent in the exponential dependence diverges as a param-eter approaches a critical value. So far this type of transientchaos has been illustrated exclusively in the phase spaceof dynamical systems. Here we report the phenomenonof noise-induced superpersistent transients in the physicalspace and explain the associated scaling law based on thesolutions to a class of stochastic differential equations. Thecontext of our study is advective dynamics of inertial parti-cles in open chaotic flows. Our finding makes direct exper-imental observation of superpersistent chaotic transientsfeasible and it also has implications to problems of currentconcern such as the transport and trapping of chemicallyor biologically active particles in large scale flows.

Younghae DoArizona State [email protected]

164 DS05 Abstracts


MS80

Finite-Scale Hamiltonian Chaotic Scattering

An adequate characterization of the dynamics of Hamilto-nian systems at physically relevant scales has been largelylacking. Here we investigate this fundamental issue usingas a paradigm the problem of Hamiltonian chaotic scatter-ing. We show that the finite-scale Hamiltonian dynamicsis governed by effective dynamical invariants, which aresignificantly different from the dynamical invariants thatdescribe the asymptotic Hamiltonian dynamics. The effec-tive invariants depend both on the scale of resolution andthe region of the phase space under consideration, and theyare naturally interpreted within a framework in which thenonhyperbolic dynamics of the Hamiltonian system is mod-eled as a chain of hyperbolic systems. Our results are il-lustrated with applications to chemical reactions in chaoticfluid flows.

Adilson E. MotterLos Alamos National [email protected]

MS80

Collision Dynamics of Raindrops and TransientChaos

The collision dynamics for an ensemble of raindrops withdistributed mass is governed by the transient (regular orchaotic) dynamics of each raindrop between successive col-lisions, when the mean collision time is relatively short.However, the dynamics of each raindrop is different due tothe difference in mass, and the traditional approach for theproblem of chaotic scattering cannot be applied directly.We study this problem by extending the ideas from thetraditional approach.

Takashi NishikawaSouthern Methodist [email protected]

MS80

Statistical Properties of Wave Chaotic Scatteringand Impedance Matrices

Edward OttUniversity of [email protected]

Xing Zheng, James Hart, Sameer Hemmady, Steve AnlageUniversity of [email protected], [email protected],[email protected], [email protected]

Thomas AntonsenInstitute for Research in Electronics and Applied PhysicsUniversity of [email protected]

MS80

Reactions in Chaotically Time-Dependent Flows

The dynamics of chemically or biologically active particlesis studied when they are advected by smooth open flows ofchaotic time dependence. Such an advection dynamics canbe modeled by a random time-dependence of the parame-ters on a stroboscopic map. A general theory is developedfor reactions in such random flows, and a reaction equationis derived. We show that there is a singular enhancement ofthe reaction, which depends only on the fractal dimensionof the filaments where particles accumulate.

Alessandro P.S. de MouraUniversidade de Sao Paulo05315-970, Sao Paulo, S.P. [email protected]



Tamas TelEotvos Lorand University, HungaryInst for Theoretical [email protected]

MS80

Long Chaotic Transients in Couette Flow

Joe Skufca and I investigate a 9 dimensional model of Cou-ette flow for Raleigh numbers where there is a chaotic sad-dle of dimension about 5. The set of points attracted to itis above 8.99. We describe this set and how it changes asRaleigh number varies.

James YorkeUniversity of [email protected]

MS81

Locomotion by Flapping as a Bifurcation

Stephen ChildressCourant Institute of Mathematical [email protected]

MS81

Wake Structures and Thrust Produced by Un-steady Finite Aspect Ratio Propulsors

Anguilliform and rajiform locomotion are studied experi-mentally using unsteady mechanical models. These modelsconsist of pitching panels, and undulating bodies represent-ing manta rays and eels. Despite significant differences intheir geometries and kinematics, these models produce sim-ilar wake structures which evolve in a manner that is dis-tinct from wakes produced by nominally two-dimensionalgeometries. Details of these wake structures will be dis-cussed as well as implications for thrust production and

DS05 Abstracts 165

propulsive efficiency.

Alexander SmitsPrinceton [email protected]

James H. J. BuchholzDepartment of Mechanical & Aerospace EngineeringPrinceton [email protected]

Richard ClarkPrinceton [email protected]

MS81

Reduced Models for Fishlike Swimmers

We study the dynamics of an articulated body in a perfectfluid. The goal of this study is to analyze the locomo-tion of the body due to the coupling between its shapechanges and the surrounding fluid dynamics. In this talk,we present two models: the first treats motion in potentialflow and the second analyzes the interaction of the bodywith point vortices. The model of the motion in poten-tial flow relies on geometric mechanics for systems withsymmetry combined with numerical simulation. The artic-ulated body model achieves forward locomotion and turn-ing maneuvers by merely changing its shape. We addressthe problem of motion planning or trajectory design as onein optimal control; that is, we seek the most efficient shapechanges that achieve a desired net locomotion. We thenintroduce point vortices in the above flow and study theirinteraction with the articulated body. We need to add tothis framework a mechanism that emulates vortex shed-ding in order to describe the dynamics of the body withself-generated vortices.

Clancy W. RowleyPrinceton [email protected]

Eva KansoCalifornia Institute of [email protected]

Juan Melli-HuberMechanical and Aerospace EngineeringPrinceton [email protected]

MS81

Lagrangian Mechanics and Thrust GenerationThrough Vortex Shedding

The locomotion of marine animals often involves the devel-opment of thrust through the shedding of vortex structures.Vortex shedding is an essentially viscous phenomenon, butcan be modeled in the context of Lagrangian mechanicsthrough the introduction of constraints generalizing Kuttaconditions. Fluid vorticity and body momentum evolve ac-cording to coupled equations with thrust forces contribut-ing gyroscopic terms to the equations governing body mo-tion. We present basic elements of this approach to mod-eling such systems.

Scott KellyDepartment of Mechanical & Industrial EngineeringUniversity of Illinois at Urbana-Champaign

[email protected]

MS81

The Lamprey as Elastic Rod: A Simple Model ofLocomotion

We will consider the locomotion of anguilliform swimmerssuch as the lamprey, eels and certain aquatic snakes. Mod-els for this type of swimming have been proposed by ei-ther specifying the shape of the swimmer or modelling theswimmer as a discrete chain of stiff links. We consider acontinuum model that incorporates the muscle activity asa change in the intrinsic curvature of an elastic rod. Wewill see how a wave of curvature passing through the rodcreates a locomoter force propelling the rod through thewater.

Tyler McMillenPrinceton [email protected]

MS81

Control Design and Flow Characterization For Op-timal Foil Thrust

In this presentation, existing models for carangiform lo-comotion are extended to include point vortices. Optimalcontrol results for a class of second order nonholonomic sys-tems are used to motivate studies of frequency-modulatedgait selection for this class of systems. Model and controlpredictions are validated using digital partical image ve-locimetry with both a pitching and heaving foil with anda free-swimming fish robot.

Kristi MorgansenDept. of Aero and AstroU. of [email protected]

MS82

Optimal Drug Infusion Strategies for CancerChronotherapy

We present opti-misation procedures for cancer chronochemotherapy. Sixcoupled differential equations govern the evolution of boththe tumour cell population and the jejunal villi population,to be shielded from unwanted drug side effects. Maximumtumour cell kill is the objective function, tolerability asmeasured by jejunal mucosa preservation the constraint.Optimisation is led with respect to cytotoxic drug infusionflow and takes into account circadian variations in drugsensitivity for both targets, healthy and tumoral.

Jean [email protected]

Claude BasdevantUniversite Paris-NordVilletaneuse, [email protected]

Francis LeviINSERM E 0354Hopital Paul-Brousse, Villejuif, [email protected]

166 DS05 Abstracts

MS82

Modelling the Malignant Progression of Brain Tu-mours

Tumours arising from the brain glia cells constitute themost common form of primary brain tumours in adults.Astrocytomas are classified by the degree of malignant ap-pearance in the pathological examination into four grades.We have developed a detailed PDE model incorporatingcellular mutation to simulate the development of a tumourfrom low to high grade. We use our model to gain insightinto the mechanistic basis of tumour progression.

Eliezer ShochatWeizmann Institute [email protected]

Kevin PainterHariot Watt University, EdinburghDepartment of [email protected]

Nick SavillDept of [email protected]

MS82

Modeling Tumor Growth Under Sub-MaximalDensities

A model describing the competition between normal andmalignant cells on nutrients under sub-maximal densitiesin the presence of angiogenesis and growth factors is de-veloped. The natural model corresponds to a semi-linearslow-fast ODE system. The properties of this model andtheir dependence on the form of the interaction terms arediscussed. For a large class of such systems, the growthfactors may be used to locally stabilize the healthy tissuesolution.


Vered Rom-KedarThe Weizmann InstituteApplied Math & Computer [email protected]

MS82

G-CSF Effect on White Blood Cells Dynamics

Prediction and prevention of infectious complications fol-lowing chemotherapy induced suppression of white bloodcells (granulocytopenia) is a necessity in oncological prac-tice. G-CSF (granulocyte colony stimulating factor) is piv-otal in therapy of patients with clinical episodes of granulo-cytopenia. We propose that important features of G-CSFeffect on granulocyte dynamics may be captured by a 2Dmodel. The low number of required parameters simplifiesthe prediction of granulocytopenia in individual patients.



Lee SegelWeizmann Institute of [email protected]

MS83

Mixing and Chemical Reactions in Chaotic Flows

Theory of fast binary chemical reaction, A + B → C, in astatistically stationary bounded chaotic flow at large Pecletnumber Pe and large Damkohler number Da is described.For stoichiometric condition we identify subsequent stagesof the chemical reaction. The first stage correspondent toformation of the developed lamellar structure in the bulkpart of the flow is terminated by an exponential decay, ∝exp(−λt) (where λ is the Lyapunov exponent of the flow),of the chemicals in the bulk. The second and the thirdstages are due to the chemicals remaining in the boundaryregion. During the second stage the amounts of A andB decay ∝ 1/

√t, whereas the decay law during the third

stage is exponential, ∝ exp(−γt), where γ ∼ λ/√Pe. We

also discuss recent experiments confirming the theory.

V. LebedevLandau Institute, [email protected]

Misha ChertkovT-13, Theoretical DivisionLos Alamos National [email protected]

MS83

Front Dynamics in Reaction-Diffusion Systemswith Anomalous Diffusion

A growing number of studies have pointed out the presenceof anomalous diffusion due to chaotic advection, and thereis a need to understand reactive systems in the presenceof this type of non-Gaussian diffusion. Here we considerfront dynamics in reaction-diffusion systems with anoma-lous diffusion due to asymmetric Levy flights. Numericaland analytical studies of the fractional Fisher-Kolmogorovequation show exponential acceleration of fronts and uni-versal power law decay of the front’s tail.

Benjamin Carreras, Vickie Lynch,Diego Del-Castillo-NegreteOak Ridge National [email protected], [email protected],[email protected]

MS83

Oscillatory Chemical Reactions in Chaotic Flows

We investigate the effect of chaotic fluid mixing on oscilla-tory and excitable chemical reactions. Mixing may quenchthe oscillations or produce coherent synchronized oscilla-tions in nonuniform oscillatory media and the amplitudeof the oscillations depends on the stirring rate. Coherentoscillations are also possible in excitable systems subjectto stochastic perturbation. In this case the stirring ratecontrols the period of the oscillations. The relationship toexperimental observations in chemical reactors will also be

DS05 Abstracts 167

discussed.

Zoltan NeufeldLos Alamos National [email protected]

Changsong ZhouPotsdam University, [email protected]

Istvan KissDepartment of ZoologyUniversity of [email protected]

Jurgen KurthsInstitute of PhysicsUnivesrity of [email protected]

MS83

Rock-Scissors-Paper Game in a Chaotic Flow: TheEffect of Dispersion on the Cyclic Competition ofMicroorganisms

Numerical simulations and experiments have shown thatthe outcome of cyclic competition is affected by the spatialdistribution of the competitors. Short range interactionand limited dispersion allows for coexistence of differentspecies whose competition would destroy the species di-versity in a well mixed environment. We study the inter-mediate situations of imperfect mixing, typical in aquaticmedia, and the transition between the two regimes, in amodel of cyclic competition between toxin producing, sen-sitive and resistant phenotypes. It is found, that chaoticmixing, by changing the structure of the spatial distribu-tion, induces oscillations in the populations. The coherenceof the oscillations increases with the strength of mixing,leading to the extinction of some species beyond a criticalmixing rate. When mixing is non-uniform in space (e.g.there are obstacles or vortices in the flow), coexistence canbe sustained at much stronger mixing, by the formation ofpartially isolated regions, that prevent global extinction.The heterogeneity of mixing may enable toxin producingand sensitive strains to coexist for very long time at strongmixing.

Zoltan NeufeldLos Alamos National [email protected]


Istvan ScheuringDept. of Plant Taxonomy and EcologyEotvos [email protected]

MS83

Experimental Studies of Front Propagation in aReaction-Advection-Diffusion System

We present experiments on propagation of chemical frontsin an alternating vortex chain that can oscillate and/ordrift laterally. For the oscillating vortex chain (with diffu-

sive mixing), the fronts mode-lock to the external forcing.We have mapped out Arnold tongues associated with thislocking behavior. We also investigate the behavior of thefronts in the regime in which transport is superdiffusive,concentrating both on the shape of the advancing frontand on its propagation speed.

Matt Paoletti, Tom SolomonBucknell [email protected], [email protected]

MS83

Effects of Particle Inertia in Active Chaos: En-hancement of Kinetics

We investigate the reaction kinetics of small spherical parti-cles with inertia, being advected by hydrodynamical flowswith imperfect mixing properties. In contrast to passivetracers, the particle dynamics is governed by the stronglynonlinear Maxey-Riley equations, which typically createchaos in the spatial component of the particle dynamics,appearing s filamental structures in the distribution of thereactants. The emerging nonlinear effects include tracertrapping and kinetics acceleration.

Izabella Benczik, Adilson Enio MotterMax-Planck-Institut fur Physik komplexer SystemeDresden, [email protected],[email protected]

Takashi NishikawaSouthern Methodist [email protected]

Tamas TelInstitute for Theoretical PhysicsEtovos University, [email protected]

Z. ToroczkaiLos Alamos National [email protected]


MS84

Stability of Nonlinear Defect Modes in Bragg Grat-ing Fibers

We consider light confined by a defect in a Bragg grat-ing optical fiber that supports multiple bound states.This system has no energy-minimization principle, but isseen in numerical experiments to select a sort of groundstate. We present results of numerical and and analyticalstudy, including eigenvalues computed numerically usingEvans functions, to discuss the stability of these compet-ing modes.

Roy H. GoodmanNew Jersey Institute of TechnologyDepartment of Mathematical [email protected]

Michael I. Weinstein

168 DS05 Abstracts

Columbia UniversityDept of Applied Physics & Applied [email protected]

MS84

Vector Soliton Interactions in Birefringent OpticalFibers

We derive a simplified system of Hamiltonian ordinary dif-ferential equations (ODEs) from ones derived by Ueda andKath for a coupled pair of nonlinear Schrodinger equationsalso studied by Tan and Yang. Using matched asymp-totic expansions for separatrix crossing, we determine ”res-onance windows” of initial conditions for which the inter-acting solitons are reflected after two passes. Numericalsimulations of these two ODE models agree quite well withour asymptotic theory of capture and resonance.

Roy H. GoodmanNew Jersey Institute of TechnologyDepartment of Mathematical [email protected]

Richard HabermanSouthern Methodist UniversityDepartment of [email protected]

MS84

Localized Structures in Two-Dimensional PhotonicLattices

New models describing wave propagation in transverselymodulated optically induced waveguide arrays are pro-posed. In the weakly guided regime, a discrete nonlinearSchrodinger equation with the addition of bulk diffractionterm and an external “optical trap” is derived. In the de-focusing regime the optical trap induces a stable localizedmode. In the limit of strong transverse guidance, the dy-namics is governed by a model which represents the opticalanalogue of wave action.

Ziad MusslimaniUniversity of Central FloridaDepartment of [email protected]

Keith A. JulienDepartment of Applied MathematicsUniversity of Colorado at [email protected]

Mark AblowitzDept. of Applied MathematicsUniversity of [email protected]

Michael I. WeinsteinColumbia UniversityDept of Applied Physics & Applied [email protected]

MS84

Modeling of Wave Resonances in Low-ContrastPphotonic Crystals

Coupled-mode equations are derived from Maxwell equa-tions for modeling of low-contrast cubic-lattice photonic

crystals in three spatial dimensions. Coupled-mode equa-tions describe resonantly interacting Bloch waves in stopbands of the photonic crystal. We study the linearboundary-value problem for stationary transmission of fourcounter-propagating and two oblique waves on the plane.Well-posedness of the boundary-value problem is provedby using the method of separation of variables and gener-alized Fourier series. For applications in photonic optics,we compute integral invariants for transmission, reflectionand diffraction of resonant waves.


MS84

Anomalous Scattering Behavior Near Guided Elec-tromagnetic Modes

Guided electromagnetic modes in periodic slabs or pillarsare responsible for frequency-localized anomalous behaviorof energy transmission through the structures. Typically,these anomalies occur near the frequency and wave vectorof a truly bound mode that becomes leaky upon perturba-tion of these parameters or the material or geometric prop-erties of the structure. We model these phenomena usingthe spectral theory of boundary integral operators and theextended Maxwell system for lossy systems. (Joint workwith S. Venakides, D. Volkov, and A. Figotin)

Stephen P. ShipmanDepartment of MathematicsLouisiana State [email protected]

MS84

Semi-Analytical Methods to Treat Dynamic Pho-tonic Crystal Systems

Electromagnetic states associated with photonic crystalsundergoing rigid time-dependent translations in positionspace are investigated. It is demonstrated that Bloch wavevector remains a conserved quantity and that an analogueof Blochs theorem for time-dependent states can be formu-lated. For the time-dependent translations involving har-monic rigid vibrations of the photonic crystal it is shownthat inter-band transitions can be induced between dis-tinct photonic bands of a crystal, thus enabling non-linearapplications with strictly linear materials.

Maksim SkorobogatiyGenie physiqueEcole Polytechnique de [email protected]

MS85

Weakly Interacting Waves and Bumps in Synapti-cally Coupled Neural Media

We analyze the existence and stability of stationary N–pulses and traveling wave trains in a one-dimensional neu-ronal network. The network is modeled in terms of a non-local integrodifferential equation whose integral kernel rep-resents the spatial distribution of synaptic weights. We usesingular perturbation theory to derive a set of N coupledODEs for the dynamics of N weakly interacting pulses, es-tablishing a direct relationship between the explicit form ofthe interactions and the structure of the long-range synap-

DS05 Abstracts 169

tic coupling.

Paul BressloffUniversity of [email protected]

MS85

Persistent Activity and Gap Junctions

It has been noted that gap junctions are most commonbetween inhibitory interneurons in the cortex. We sug-gest that the reason that there are few observations of gapjunctions between excitatory cells is due to their abilityto disrupt ersistent activity. We show that they do thisthrough two mechanisms. First, GJs provide an decreasedmembrane resistance so that synaptic excitation is less ef-fective. Secondly, they can destabilize the asynchronousfiring state making it more difficult for maintained persis-tent activity.

Bard ErmentroutUniversity of Pittsburgh,Pittsburgh, [email protected]

MS85

An Intracellular Ca2+ Subsystem As a BiologicallyPlausible Source of Intrinsic Bistability in a Net-work Model of Working Memory

We explore a network model of working memory in an in-tegrodifferential form similar to those proposed by Amari.The model incorporates an intracellular Ca2+ subsystemwhose dynamics depend upon the second messenger [IP3].This Ca2+ subsystem endows individual units with in-trinsic bistability for a range of [IP3]. This full networksustains [IP3]-dependent persistent (bump) activity in re-sponse to a brief transient stimulus. Our results highlightthe importance of second messenger activity time scales.

Chris FallNYU Center for Neural ScienceNew York [email protected]

MS85

Some Neural Examples of “Equation-Free” Mod-elling

“Equation-free” modeling allows one to analyse the effec-tive equations governing the macroscopic dynamics of asystem, even if they cannot be explicitly derived, providingthat the microscopic dynamics can be simulated and thatthere is some separation of timescales. These techniquescan be used to do bifurcation analysis of realistic networksof neurons. We demonstrate this for macroscopic steadystates of small networks, and for spatiotemporal patternssuch as bumps and waves.

Carlo R. LaingMassey [email protected]

MS85

Wave Propagation in Networks of Electrically Cou-

pled Cortical Interneurons

Fast-spiking (FS) neurons in the cortex are connected byboth electrical coupling and inhibitory synapses. The ex-tensive electrical coupling and the high level of excitabilitymake the FS cell network well-suited to support propagatedwaves of activity. However, the thin spikes and large after-hyperpolarizations of FS cells and inhibitory connectionsbetween FS cells should work against wave propagation.We discuss conditions for the existence of waves and theproperties of these waves in models of cortical FS cell net-works.

Tim LewisUniversity of Californiaat [email protected]

MS85

Understanding Timing and Structure of CorticalWaves From Data

Waves, a fundamental mechanism for conveying informa-tion, are observed in many areas of the brain but theirstructure is not well-understood. We have developed sev-eral visualization techniques such as latency-ordered space-time overlays, reordered parallel coordinate visualizationsand subspace projections. These techniques permit de-tailed comparisons of activation for data from different tri-als or from different variables in the same trial. We showhow application of these techniques to cortical waves pro-vides insight into underlying mechanisms.

Kay A. RobbinsDept of Computer Scienc and Cajal NeuroscienceResearch CentUniversity of Texas at San [email protected]

MS86

PCR in in-vitro Diagnostics - Dynamics and DataAnalysis

This talk will describe methods for data analysis using dy-namical models and statistical methods in in-vitro diag-nostic applications of Polymerase Chain Reaction (PCR).Models and parameter spaces of PCR and related biochem-istry will be presented. Data will be presented to outlinethe scope of the data analysis problem. Methods and re-sults for determining the parameters with enough accuracyand robustness to allow for their use in diagnostic applica-tions will be described.

David Eyre, Aldo BernasconiIdaho [email protected], aldo [email protected]

MS86

Computational Cell Biology in the PharmaceuticalIndustry

In contrast to industries that have already embraced com-puter simulation in guidance of product design and dis-covery, computational biology, in particular mechanisticsimulation of cellular processes, remains a marginal (al-beit growing) activity in the pharmaceutical industry. Iwill present a few success stories – and pitfalls – frommy personal experiences applying biological simulation andmathematics to cancer drug discovery and cardiac safety

170 DS05 Abstracts

assessment. I will also emphasize the mathematical toolsand techniques I used, or learned to use, along the way.Finally, I will describe a few of the biological simulationactivities that some of the more innovative drug discoveryand biotechnology companies are currently pursuing.

Dean BottinoThe BioAnalytics [email protected]

MS86

Panel Discussion

Eric N. CytrynbaumUniversity of British ColumbiaDepartment of [email protected]

MS86

Virtual Patients in Clinical Trial Design

Virtual Patients developed by Entelos scientists providenovel methodologies for predicting clinical outcomes andoptimizing clinical trial designs. Developed within a pro-prietary simulation platform, Virtual Patients are explicitrepresentations of observed clinical phenotypes and providea biological context for predicting compound efficacy andidentifying biomarkers of therapeutic outcome. A case-study demonstrating the application of Virtual Patients tooptimize a Phase I trial design, which led to substantialtime and cost savings, will be presented.

Cynthia J. MusanteEntelos, [email protected]

MS86

Application of a Computational Fluid DynamicsModel to Predict Uptake of Hydrogen Sulfide inRat Nasal Passages

An anatomically accurate computational fluid dynamicsmodel of rat nasal passages was used to simulate inspi-ratory airflow and calculate wall mass flux of inhaledhydrogen sulfide (H2S). The reaction-diffusion equationwas solved with a nonlinear boundary condition describingelimination of H2S in nasal tissue by first-order and sat-urable pathways. Rate constants were computed from anair-tissue compartmental model that was fit to experimen-tal measurements of nasal H2S uptake in rats.

Paul M. SchlosserNCEA, U.S. [email protected]

Jeffry Schroeter, Anna McElveen, David Dorman, JuliaKimbellCIIT Centers for Health [email protected], [email protected], [email protected],[email protected]

MS87

Prediction of Chaos in a Mechanical System withFriction

We are aimed to illustrate how the classical Melnikovs tech-nique can be extended to analyse even complex lumped

mechanical systems including non-smooth behaviour likethose with Coulomb like friction. As an example serves arotated Froude pendulum harmonically excited. It is as-sumed that in a place of pendulum fixation the Coulombfriction occurs and that the pendulum pivot rotates with aconstant velocity. The classical Melnikovs technique is usedto predict chaotic behaviour in our one-degree-of-freedommechanical system. The obtained results are verified by nu-merical experiments. The expected chaotic thresholds aredrawn in the control parameter planes. In what follows forone fixed value of the first control parameter the associatedcritical value of the second control parameter is computedowing to its prediction given by the homoclinic bifurcationcondition. Then numerical experiment is carried out us-ing the first control parameter. A range of changes of thebifurcation parameter includes the predicted value of thesecond parameter owing to the applied Melnikovs method.The numerical verification relies on finding a value of thebifurcation parameter when the homoclinic bifurcation oc-curs, and on its comparison with the value predicted an-alytically. In addition, phase portraits and Poincar mapsare used for identification of chaos. It has been shown thatowing to application of the Melnikovs technique, the criti-cal values of the parameter associated with the homoclinicbifurcation have been analytically predicted properly sincethey have been also identified by the numerical compu-tations. Owing to existence in the non-perturbed systemfour homoclinic orbits (in the case of relatively high rotat-ing velocity) four analytical chaos criterions have been alsocomputed. The carried out numerical analysis fully coin-cides with the predicted bifurcations region separated by aperiodic window. 1. J. Awrejcewicz, M. M. Holicke, Mel-nikov’s Method and Stick-Slip Chaotic Oscillations in VeryWeekly Forced Mechanical systems, International Journalof Bifurcation and Chaos, 9(3), 1999, 505-518. 2. J. Awre-jcewicz, D. Sendkowski, How to predict stick-slip chaos inR4, Physics Letters A, 330, 2004, 371-376.

Mariusz HolickeTechnical University of [email protected]

Jan AwrejcewiczDepartment of Automatics and BiomechanicsLodz Technical [email protected]

MS87

Dynamics at a Degenerate Graze for An ImpactOscillator

We give some rigorous results on dynamics for an impactoscillator close to a degenerate graze. These results under-pin earlier analysis by [Budd and Dux, Phil. Trans. R. Soc.Lond. A 347 (1994), 365–389] and others on intermittencyand chattering.

David ChillingworthUniversity of SouthamptonSchool of [email protected]

MS87

Control of Near-Grazing Dynamics in Impact Os-cillators

A method is presented for controlling the persistence of alocal attractor near a grazing periodic trajectory in a piece-wise smooth dynamical system in the presence of jump dis-

DS05 Abstracts 171

continuities. Specifically, a discrete, linear feedback strat-egy is employed to sustain the existence of an attractornear the grazing trajectory, such that the deviation of theattractor from the grazing trajectory goes to zero as thesystem parameter values approach those corresponding tograzing contact.


Jenny JerrelindDepartment of Aeronautical and Vehicle EngineeringRoyal Institute of [email protected]

MS87

Co-dimension-two Grazing-sliding Bifurcations inFilippov Systems

Many systems of relevance to engineering applications aremodelled as sets of ordinary differential equations with dis-continuous right-hand sides. These systems are termed asFilippov systems and are known to exhibit so called slidingbifurcations. We will analyse two distinct codimension-twoscenarios occuring when a limit cycle exhibiting grazing-sliding is non-hyperbolic. A dry-friction oscillator wherethe aforementioned types of codimension-two events werefound will be analysed, and its dynamics around thecodimension-two points will be subsequently explained.

Piotr KowalczykDept. Engineering MathematicsUniversity of [email protected]

MS87

Chaotic Oscillations in Piecewise Linear SystemsWith Hysteresis

The proposal of this talk is to analyze the occurrence ofchaotic oscillations when a piecewise linear system is builtby applying a switching element with hysteresis in the feed-back loop. The analysis is performed for systems of secondand third orders and it is shown that chaotic oscillators ofdouble-scroll type can be obtained and that the simplicityof their structure allows to establish a systematic procedurefor the construction of chaotic oscillators.

Pedro L. D. PeresFaculty of Electrical Engineering and ComputationState University of Campinas - [email protected]

Ubirajara MorenoDepartment of Automation and SystemsFederal University of Santa Caterina - [email protected]

MS87

Grazing and Chattering in Vibro-Impact Systems

Grazing and chattering are two nonsmooth phenomenathat appear frequently in vibro-impact systems. Theseevents are also well known to generate additional prob-lems in numerical analysis. To gain some understanding ofwhen to expect such events we will examine the interplay

between grazing and chattering in a simple impacting pen-dulum. We will also show how grazing can organise thebasin of attraction for systems with coexisting limit cycles.

Chris BuddUniversity of [email protected]

Petri T. PiiroinenDepartment of Engineering MathematicsUniversity of [email protected]

MS88

Odor Coding in Olfactory Networks

With computer model of olfactory system we explored in-trinsic and synaptic mechanisms involved into coding olfac-tory information. Interaction between principle and localneurons in the antennal lobe created odor specific temporalpatterns of activity decoded by the neurons at downstreamlevels of processing. A combination of nonlinear intrinsicproperties of the downstream cells and local inhibitory cir-cuits favored coincidence detection tuned to select correla-tions in the input spike trains and promoting sparse odorrepresentation.

Maxim BazhenovSalk [email protected]

MS88

Computation with Spikes

In the neocortex, neurons project to distant regions re-sulting in conduction delays of tens of milliseconds. Syn-chronous spiking of such neurons may not be effective tofire a postsynaptic cell, since the spikes arrive to the cell atdrastically different times. To excite the cell, the neuronsmust fire with certain polychronous spiking patterns deter-mined by the delays. Simulating a spiking network with de-lays and STDP, we show that neurons spontaneously poly-chronize, i.e., self-organize into groups and fire time-lockedbut not synchronous spike-timing patterns with millisecondprecision. We found more groups than neurons resultingin unprecedented memory capacity of the system.

Eugene M. IzhikevichThe Neurosciences [email protected]

MS88

A Unified Approach to Determining EmbeddingDimension and Time Delay for Multivariate TimeSeries

Typically there are two problems in multivariate attrac-tor reconstruction. One is finding the time dely for eachtime series and the other is establishing the embedding di-mension. The latter means deciding which series to use toenable a complete unfolding of the attractor. Autocorre-lation or mutual information is used in the time-delay es-timation and false-nearest neighbor statistics are used forthe embedding dimension. We approach these two seem-ingly independent problems by returning to Taken’s em-bedding theorem and examining what the mathematicalrequirement is to add another component to an attractorreconstruction. We show that the above two problems are

172 DS05 Abstracts

really the same problem and there is a unified approach toboth that logically leads to finding correct time delays andsimultaneously choosing which time series to use.

Linda J. MonizU.S. Geological [email protected]

Louis M. PecoraNaval Research [email protected]

Thomas L. CarrollNaval Reseach [email protected]

Jon NicholsNaval Research LabWashington, DC, [email protected]

MS88

Beamforming with Repulsive Phase Synchroniza-tion in an Array of Nonlinear Oscillators

An array of oscillators with all-to-all repulsive coupling isconsidered. When the oscillators are identical, the dynam-ics of the array settle to a synchronized regime, character-ized by a zero mean field. However, for non-identical os-cillators, the mean field is non-zero for arbitrary couplingstrength. We develop a theoretical description of repulsivesynchronization based on a phase approximation of the dy-namics. We also study the mean field response of the arrayto external periodic driving. The concept can be used inconjunction with a phased array antenna to improve beam-forming characteristics.

Michael GabbayInformation Systems [email protected]

Lev S. TsimringUniversity of California, San [email protected]

Nikolai RulkovISL [email protected]

Michael L. [email protected]

MS88

Noise Controlled Oscillations in Globally CoupledElements

We demonstrate that noise can play a certain orderingrole in ensembles of globally coupled oscillators. As ex-amples the active rotators (the Kuramoto model) and theFithHugh-Nagumo excitable systems are taken. On ap-proximating the instantaneous distribution of the oscilla-tors by the Gaussian one, the Langevin equations are re-duced to dynamical systems which govern the evolution ofthe cumulants. Bifurcation analysis discloses that increaseof noise leads from stationary states through complicated

time-dependent regimes to broad stationary distributions.

Alexander NeimanDepartment of Physics and AstronomyOhio [email protected]

Lutz Schimansky-GeierInstitute for PhysicsHumboldt University at [email protected]

Michael ZaksDepartment of Stochastic Processes, Institute of PhysicsHumboldt University of [email protected]

Xaver SailerInstitute of PhysicsHumboldt University of [email protected]

MS89

Propagation of Singularities in Director Fields

We shall discuss some results on the simplest possible heatflow of director fields with values in the unit sphere. Thereturns out to be a fascinating interplay between the ex-istence of point singularities, also called defects, and thenonuniqueness of the flow caused by a certain degree offreedom in prescribing the evolution of the singularities.The singularities can be created spontaneously, startingfrom smooth data. Part of the lecture will be devoted tothe construction of traveling wave solutions, and severalopen problems will be presented.

Michiel BertschIstituto per le Applicazioni del Calcolo M. [email protected]

MS89

Phase Locking for Flame Propagation in PeriodicMedia

In some situations, flame propagation can be describedqualitatively by sharp-interface models with energy depo-sition at the interface. For propagation in periodic mediawe examine phase-locking effects (which are analogous tothe ones observed for the classical van der Pol oscillator).In some sense the free-boundary PDE system behaves likea fancy nonlinear oscillator. Amazingly, many features ofPDE dynamics are reproduces in a 3-dimensional projec-tion of the original dynamical system. We use correlationdimension maps as a tool for investigating underlying dy-namics. This is a joint work with Michael Frankel of Indi-ana University-Purdue University Indianapolis.

Victor RoytburdDepartment of Mathematical [email protected]

MS89

Signal Transmission by Autocrine Cells in ModelEpithelial Layers

Autocrine signaling induced by growth factors is crucial invarious stages of development and in adult multicellular or-ganisms across species. At the present level of complexity,

DS05 Abstracts 173

systematic evaluation of cell communication mechanismsis next to impossible without mathematical modeling ofcell signaling networks. In this talk, I will discuss recentresults of our mechanistic modeling and analysis of Epider-mal Growth Factor Receptor (EGFR)-mediated cell com-munication. Our modeling and analysis allows to charac-terize signal transmission in epithelial layers in terms ofthe biophysical and geometric parameters of the problem.

Stas ShvartsmanLewis Sigler Institute for Integrative Genomicsand Chemical Engineering Department, [email protected]

Cyrill B. MuratovNew Jersey Institute of TechnologyDepartment of Mathematical [email protected]

MS89

Surprising Aspects of Traveling Waves with Non-linear Diffusion: Bacterial Growth Vortex Diffu-sion Models

Equations with nonlinear diffusion, like in the well knownporous medium equation, arise in variety of systems. Inthis talk we will discuss two examples of reaction diffusiontype equations with nonlinear diffusion, which admit mov-ing fronts. The first case concerns a bacterial growth prob-lem, the second describes the penetration of a vortices insuperconductors. Both equations admit moving front solu-tions which exhibit some surprising behavior: the bacterialgrowth fronts do not properly map onto a moving bound-ary problem in the thin front limit, while the nonlinearvortex fronts show a regime of singular behavior which isdifferent from that expected naievely on the basis of theirsimilarity with the porous medium equation.

Wim van SaarloosInstituut-LorentzLeiden [email protected]

MS90

Patterns of Turbulence

Plane Couette flow – the flow between two infinite parallelplates moving in opposite directions – undergoes a discon-tinuous transition from laminar flow to turbulence as theReynolds number is increased. Due to its simplicity, thisflow has long served as one of the canonical examples forunderstanding shear turbulence and the subcritical tran-sition process typical of channel and pipe flows. Only re-cently was it discovered in very large aspect ratio experi-ments that this flow also exhibits remarkable pattern for-mation near transition. Steady, spatially periodic patternsof distinct regions of turbulent and laminar flow emergesspontaneously from uniform turbulence as the Reynoldsnumber is decreased. The length scale of these patternsis more than an order of magnitude larger than the plateseparation. It now appears that turbulent-laminar patternsare inevitable intermediate states on the route from turbu-lent to laminar flow in many shear flows. I will explainhow we have overcome the difficulty of simulating theselarge scale patterns and show results from studies of threetypes of patterns: periodic, localized, and intermittent.

Dwight Barkley

University of WarwickMathematics [email protected]

Laurette S. [email protected]

MS90

Strange Saddles in Pipe Flow

The fact that turbulent trajectories in the transition regioncan decay without prior indication and without the influ-ence of noise suggests that the turbulent state cannot bean permanently sustained attractor. The most likely phasespace structure then is that of a chaotic saddle: they arecharacterized by an exponential distribution of life timesand a positive Lyapunov exponent for the motion close tothe saddle. There is experimental evidence for such sad-dles in plane Couette flow and numerical evidence in planeCouette, pipe flow and several models.


MS90

Patterns of Vortices and Jet Streams in Rotating,Stratified Shear Flows

Rotating, stratified flows are nearly 2D and the inverse cas-cade of energy often leads to large, turbulent vortices andjets. In general, the flows are not unique, and there areseveral basins of attraction of the flow - each character-ized by its own pattern of vortices and jet streams. Thetransport properties of each pattern vary markedly, so ina geophysical, or climate-change context, the robustness ofeach pattern and how patterns are selected due to smallchanges in the environment are important. We explorepattern selection using numerical simulation and statisti-cal mechanics.

Philip MarcusUniversity of California at [email protected]

MS90

Complex Behavior of Large Scale Features of FullyTurbulent Flows

It is commonly believed that the very large level of noisearising from small-scale structures of turbulent flows pre-cludes their large scale features from displaying complexbehaviour. At variance with this widely accepted view,we will present recent experimental results obtained in twodifferent flow configurations which show that non-triviallarge-scale behaviour can persist even when considerablesmall-scale activity is present.

Louis MarieLaboratoire de Physique des OceansUniversite de Bretagne [email protected]

MS91

Verification of Hyperbolicity and

174 DS05 Abstracts

Non-Hyperbolicity Via Topological Methods

In this talk, we propose two rigorous computational meth-ods, one for detecting homoclinic tangencies, and the otherfor proving hyperbolicity of invariant sets. These methodsare combinations of several tools and algorithms, includ-ing the interval arithmetic, the subdivision algorithm, theConley index theory, and the computational homology the-ory. Applying these methods, we investigate the structureof the parameter space of the Henon family.

Zin AraiDepartment of MathematicsKyoto [email protected]

MS91

Covering Invariant Manifolds with Fat Trajectories

An invariant manifold of a flow can be defined as the imageof a smooth curve of initial points under the flow. To com-pute the stable and unstable manifolds of a fixed point,for example, a small sphere is used in the appropriateeigenspace of the linearize flow. These manifolds can bequite complicated, and the computational problem is es-sentially a mesh generation problem. We use an approachused previously for implicitly defined manifolds that repre-sents the manifolds as the union of a set of spherical neigh-borhoods, which is the geometric dual of the tiling usedin mesh based methods. This avoids the many difficultieswith advancing a mesh, but require a method of finding aneighborhood of a point on the invariant manifold. Nearthe starting curve it is straightforward to find such a neigh-borhood. The key result we use is a set of equations forthe evolution of the neighborhood along a trajectory (afat trajectory). Since trajectories may diverge, we also de-rive a result that ensures the existance of an interpolationpoint in the cleft of the diverging trajectories. The resultis a covering of the manifold, with neighborhoods of pointswhich either lie on a trajectory or are interpolated fromthe interior of a simplex whose vertices lie on trajectories.The method based on these results works for any dimen-sions (of the manifold and the embedding space), thoughlarge pieces of manifolds of dimension larger than four areprobably too expensive to compute. We demonstrate thealgorithm on the stable manifold of the origin in the Lorenzequations.

Michael E. HendersonIBM ResearchTJ Watson Research [email protected]

MS91

Rigorous Numerics for Continuation Methods inInfinite Dimension

In this talk, we will introduce a rigorous numerical methodto continue the equilibria of dissipative parameter depen-dent partial differential equations. This method is basedon a standard finite dimensional path-following algorithmadapted to the infinite dimensional case. We will presenta theorem of existence and uniqueness for the equilibriabranches of the original PDE and discuss the computa-tional complexity of the method.

Jean-Philippe LessardGeorgia Institute of [email protected]


MS91

A Numerical Method for Proving Hyperbolicity ofComplex Henon Mappings

We describe a computer program to establish whether forgiven a,c, the complex Henon diffeomorphism, H : C2 →C2, given by H(x, y) = (x2 + c − ay, x), is hyperbolic. AHenon map is hyperbolic if over each point in its chain re-current set, there is a splitting of the tangent bundle intotwo directions, which are uniformly expanded/contractedby the map. Hyperbolic maps display interesting dynam-ical behavior, and are amenable to analysis. We will alsodiscuss several examples of program output.

Suzanne Lynch HruskaIndiana [email protected]

MS92

Nonlinear Dynamics and Instabilities of CoupledMicrobeam Arrays for Scanning Probe Microscopy

Oded GottliebFaculty of Mechanical EngineeringTechnion, Haifa, [email protected]

MS92

Dynamics of Nonlinear Coupled MEMS Res-onators

We are studying the dynamics of nonlinear coupled os-cillators, motivated by recent experiments with arrays ofMEMS resonators at Caltech and Cornell. Our studies todate have focused on the weakly nonlinear regime wherewe have looked at (1) The response of 1-dimensional ar-rays of coupled oscillators to parametric excitation; and(2) The synchronization of coupled mechanical resonatorswith a distribution of frequencies. We have obtained exactresults for the parametric excitation of small arrays usingsecular perturbation theory [1], as well as an amplitudeequation to describe the slow dynamics of the parametricexcitation of large arrays [2], with many features in com-mon with Faraday waves. We have investigated a model ofsynchronization, based on reactive coupling and nonlinearfrequency pulling [3] (rather than the more common lineardissipative models), obtaining a phase diagram for the on-set of synchronization, exhibiting interesting hysteretic be-havior. [1] PRB 67 (2003) 134302; [2] cond-mat/0411008;[3] PRL 93 (2004) 224101.

Ron LifshitzSchool of Physics and AstronomyTel Aviv [email protected]

Michael C. CrossDepartment of PhysicsCalifornia Institute of [email protected]

DS05 Abstracts 175

MS92

The Stochastic Dynamics of Micron and SubmicronScale Mechanical Oscillators

The stochastic dynamics of micron and submicron scaleoscillators of arbitrary geometry immersed in a viscousfluid will be discussed. It will be shown that by using thefluctuation-dissipation theorem it is possible to calculate,using deterministic calculations, the stochastic dynamicsthat would be measured in experiment. This approach isused to investigate the motion of single and multiple os-cillators for a variety of experimentally realistic geometriesfor use in making single molecule measurements.

Mark PaulDepartment of Mechanical EngineeringVirginia [email protected]

MS92

Sensors Utilizing Parametric Resonance

Kimberly TurnerDept Mech Env EngUC Santa [email protected]

MS93

Complex Intermittent Dynamics in Population Bi-ology: A Tale of Weak Transverse Stability

New dynamical scenarios with extreme complex behaviorscan help us to understand the dynamical complexity of eco-logical systems. One of these scenarios leads to a type ofintermittence, named on-off intermittency that is relatedto the presence of an invariant subspace and to its weaktransversal unstability. In the dynamics of a communitythis subspace can be defined by the extinction of a rarespecies. The weak invading capacities of this species areresponsible of the intermittent dynamics, owing to speciescompetitions, species interactions and/or to the fluctua-tions of the environmental capacities. This kind of inter-mittent dynamics could explain the extreme fluctuationsseen on marine communities.

Bernard CazellesCNRS UMR 7625Ecole Normale [email protected]

MS93

The Dynamical Detective: Using Nonlinear Modelsto Test Ecological Hypotheses

Roughly one in three animal populations exhibits some-what cyclic dynamics. A plethora of explanations existsbut we rarely know which applies to a given population:empirical tests are usually infeasible, or inconclusive be-cause multiple mechanisms are operative. I will describecase studies where statistically rigorous comparisons ofprocess-based nonlinear models have resolved longstand-ing questions or overturned established explanations forlong-term cycles, and a laboratory model system wherethis approach has been validated experimentally.

Steven EllnerDepartment of Ecology and Evolutionary BiologyCornell University

[email protected]

MS93

Linking Mechanistic Models with EpidemiologicalData: Parameter Estimation in the Face of Incom-plete Information

The connection of ecological time-series data with mecha-nistic models poses severe statistical challenges. Typically,the dynamical models themselves are stochastic and non-linear, only a few state variables can be observed, and theseonly with error. To deal with these problems, researchershave often been forced to incorporate unpalatable assump-tions in their models for the sake of the statistics. Wedescribe the method of simulated composite likelihood, ageneral approach that requires no such assumptions. Weapply the method to measles, whooping cough, and choleraincidence data, using mechanistic continuous-time modelsfor transmission dynamics to obtain insight into the dy-namics of epidemics.

Subhash LeleStatisticsUniversity of [email protected]

Pejman RohaniInstitute of EcologyUniversity of [email protected]

Aaron A. KingUniversity of TennesseeEcology and Evolutionary [email protected]

Mercedes PascualDept. of Ecology and Evolutionary BiologyUniversity of [email protected]

MS93

Panel Discussion

James Nichols, Linda J. MonizU.S. Geological Surveyjim [email protected], [email protected]

MS95

Solutions of the NLS-Whitham Equations and Gen-eration of Intense Optical Pulses

Gino BiondiniState University of New York at BuffaloDepartment of [email protected]

MS95

Asymptotic Stability of Ground States in NLS

My talk will focus on the stability of nonlinear boundstatesthat bifurcate from the linear ones. Variational techniqueshave been developed to study their orbital stability. Morerecently, center manifold methods have been used to showtheir aymptotic stability in supercritical regimes, for ex-

176 DS05 Abstracts

ample cubic NLS in 3-d. I will present a refinement ofthe latter that includes results for the critical regimes, forexample cubic NLS in 2-d.

Eduard KirrUniversity of [email protected]

MS95

Thermally Induced Dynamics and Pattern Forma-tion in Optical Parametric Oscillators

Optical parametric oscillators (OPOs) are an importantsource of laser-quality light in the far infrared. When op-erated at high average powers, absorption can lead to sig-nificant heating of the gain medium, changing the cavityproperties and leading to thermal lensing and deformationof the transverse beam profile. To understand this processbetter, we consider the formation and evolution of patternsin reduced models of OPOs coupled to a diffusion equationfor the temperature.

Richard O. MooreNew Jersey Institute of [email protected]

MS95

Persistence and Stability of Discrete Vortices inDiscrete NLS


MS96

ComputingOne-Dimensional Manifolds of Poincare Maps inSlow-Fast Systems

We present an algorithm to compute the one-dimensionalstable and unstable manifolds of a saddle-periodic orbit ona Poincare section by setting up the problem as a boundaryvalue problem. The start point of the orbit is varied alonga piece of manifold previously computed, such that the endpoint traces out a new section of manifold. In this way, wecompute manifolds for a slow-fast chemical oscillator forthe first time.



James P. EnglandDepartment of Engineering MathematicsUniversity of [email protected]

MS96

Computing Canards in Relaxation Oscillations

Relaxation oscillations are periodic orbits in singularly per-

turbed systems that contain both slow and fast segments.Canards are trajectory segments that lie on an unstableslow manifold. Because of their instability, it is almostimpossible to compute canards by solution of initial valueproblems forward in time. We explain this more fully withillustrations and describe the use of boundary value meth-ods to compute periodic orbits containing canards.


MS96

Slow-Fast Time Scales and the Phase-Resetting Re-sponse of Cardiac Oscillators

Injection of a brief stimulus pulse phase-resets the spon-taneous periodic activity of cardiac pacemaker celss: anearlier stimulus generally delays the next action potential,while a later one causes an advance. In a slow-fast systemthe transition from delay to advance can appear discon-tinuous, as seen in experiments on some cardiac prepara-tions. Using continuation methods in AUTO, we show thatthe apparent discontinuity occurs when stimuli transportstate-points to either side of the stable manifold of the weakunstable manifold.

Leon GlassMcGill UniversityDepartment of [email protected]

Michael R. GuevaraDepartment of PhysiologyMcGill [email protected]

Eusebius J. DoedelDepartment of Computer ScienceConcordia [email protected]

Trine Krogh-MadsenMcGill [email protected]

MS96

A Unifying Framework for Reduction Methods

The identification of low-dimensional manifolds containingthe essential dynamics of multiscale systems is essential tothe construction of reduced models that can be simulatedefficiently. Here, we examine two reduction techniques, theZero-derivative Principle of Gear and Kevrekidis and theComputational Singular Perturbation of Lam and Gous-sis. We review briefly their approximation properties andconstruct a unifying framework by showing that the tech-niques’ successive iterations generate new state space co-ordinates similar to the Fenichel coordinates.

Antonios ZagarisBoston [email protected]

MS97

Extreme Ship Motion Prediction Capabilities of

DS05 Abstracts 177

the US Navy

Ships of the U.S. Navy must not only survive in large seas,but also continue to operate effectively. Predictions of largeamplitude motions and loads are attained through a mix ofmodel testing and numerical simulations which must be ofsufficient fidelity to capture the dominant nonlinear forces.While model testing will never be replaced, there is in-creased confidence in the ability to use simulation tools toassess maximum wave loads and dynamic stability perfor-mance.

William BelknapSeakeeping Division of Hydromechanics DepartmentNaval Surface Warfare Center Carderock [email protected]

MS97

On the Practical Ergodicity of Parametric Rolling

Experimental and numerical results regarding the problemof ergodicity of parametric roll in longitudinal irregularwaves indicate that temporal averages can be associatedto very large coefficients of variation, even though theirexpected values are theoretically correct estimators of en-semble averages. It has been shown that, in some cases, theanalysis of time series of typical length (30min full scale)could be useless if carried out on a single realization.

Alberto FrancescuttoUniversity of Trieste, Italytba

Gabriele BulianNaval Architecture, Ocean and EnvironmentalEngineeringUniversity of Trieste, [email protected]

Claudio LugniINSEANItalian Ship Model Basin, [email protected]

MS97

Nonlinear Ship Rolling Motion and Capsizing inRandom Waves

TBD

Jeffrey FalzaranoSchool of Naval Architecture and Marine EngineeringUniversity of New [email protected]

MS97

Ship Instabilities Due to the Asymmetric Surgingof Ships in a Following Seaway

The strongly nonlinear response of a ship in a followingseaway where it could be performing large-amplitude lon-gitudinal oscillations around a mean forward speed will bediscussed. An improved mathematical model of ship surg-ing in steep following wave has been developed. Analyticalprediction formulae for the higher limit of asymmetric surg-ing (threshold of global surf-riding) will be reported. Alsowill be discussed the role of asymmetric surging for broach-

ing and for capsize in the neighborhood of wave crests.

Kostas SpyrouSchool of Naval Architecture and Marine EngineeringNational Technical University of [email protected]

MS98

Attractors for Stochastic Lattice Dynamical Sys-tems

We consider a one-dimensional lattice with diffusive near-est neighbor interaction, a dissipative nonlinear reactionterm and additive independent white noise at each node.We prove the existence of a compact global random at-tractor within the set of tempered random bounded sets.An interesting feature of this is that, even though the spa-tial domain is unbounded and the solution operator is notsmoothing or compact, pulled back bounded sets of initialdata converge under the forward flow to a random com-pact invariant set. This is joint work with Kening Lu andHannelore Lesei.

Peter BatesMichigan State UniversityDepartment of [email protected]

MS98

Modulation Equations and Stochastic Bifurcationsfor Large Domains

We consider a class of SPDEs (e.g. stochastic Swift-Hohenberg equation) on large domains. Near its change ofstability we rigorously verify, under the appropriate scal-ing, that solutions can be approximated by a periodic wave,which is slowly modulated by the solutions to a complexstochastic Ginzburg-Landau equation. This approxima-tion also extends to invariant measures of these equations.J.w.w. M.Hairer (Warwick) and G.Pavliotis (Imperial).

Dirk BlomkerRWTH [email protected]

MS98

Pullback Attractors for Asymptotically CompactNonautonomous/Random Dynamical Systems

We introduce the concept of pullback asymptotically com-pact non-autonomous/random dynamical system as an ex-tension of the similar concept in the autonomous frame-work, and prove a result ensuring the existence of a pull-back attractor for a non-autonomous/random dynamicalsystem under the general assumptions of pullback asymp-totic compactness and the existence of a pullback absorbingfamily of sets. Finally, we illustrate the theory with a 2DNavier-Stokes model in an unbounded domain.

Tomas CaraballoDepartamento de Ecuaciones Diferenciales y AnalisisNumericoUniversity of Sevilla, 41080 Sevil, [email protected]

MS98

Smooth Conjugacy for Random Dynamical Sys-

178 DS05 Abstracts

tems

The study of reducing a nonlinear deterministic dynami-cal system to the simplest possible form (called a normalform) goes back to Poincare and Birkhoff. In this talk, Iwill report our recent work on smooth conjugacy for ran-dom dynamical systems. Our results include extensions ofPoincare and Sternberg’s theorems in deterministic casesto random dynamical systems, which are based on the Lya-punov exponents. Random dynamical systems arise in themodeling of many phenomena in physics, biology, climatol-ogy, economics, etc. when uncertainties or random influ-ences, called noises, are taken into account. These randomeffects are not only introduced to compensate for the de-fects in some deterministic models, but also are often ratherintrinsic phenomena. This is joint work with W. Li.

Kening LuBrigham Young [email protected]

MS99

Synchronization Measures in Seizure Predictionand Detection

We present results of applying various measures of dy-namical system synchronization, including new measuresderived from the Intrinsic Timescale Decomposition, tolong ECoG recordings. These measures are compared withother nonlinear dynamical measures, results from real-timeautomated seizure detection algorithm outputs, and expertvisual analysis. Implications for seizure prediction algo-rithms using these measures will be discussed.

Mark G. FreiFlint Hills Scientific, [email protected]


Thomas PetersFlint Hills Scientific, [email protected]

Ivan OsorioUniversity of KansasFlint Hills Scientific, [email protected]

Mary Ann HarrisonInstitute for Scientific Research, [email protected]

MS99

Dynamical Evolution of Seizures

We developed a novel geometrical interpretation and nu-merical approach to canonical multivariate linear discrim-ination of Fisher (1936). We found significant extractionof unique initial, middle, and terminal phases from 21 of24 scalp and intracranial recordings using discriminationof dynamical measures. No consistent increased synchro-nization was evident within the initial or terminal phasesof these seizures. These results argue for an evolution ofseizure patterns that can be partitioned based on dynami-

cal measures.

Rohit KumarThomas Jefferson High School of Science and [email protected]

Steven WeinsteinChildren’s National Medical [email protected]

Tim SauerDepartment of MathematicsGeorge Mason [email protected]

Steven J. SchiffGeorge Mason UniversityThe Krasnow [email protected]

MS99

Are Seizures Temporally Interdependent?

The effect seizures may have on subsequent seizures hasreceived little attention. In an experimental (rat) model,short (long) seizures followed short (long) interictal peri-ods with statistically significant probability. Non-randomseizure recurrence in humans was investigated using pro-longed ECoG; the results were compared with randomizedsurrogate data. Significant nonlinear dependencies of theinterseizure interval were found (p < 0.05) in over half of24 subjects. Similar outcomes were obtained for the hourlyseizure frequency and measures of seizure severity.Supported by NINDS/NIH Grant NS046060-01.

Sridhar Sunderam, Mark FreiFlint Hills [email protected], [email protected]

Ivan OsorioUniversity of KansasFlint Hills Scientific, [email protected]

MS99

Do Changes in the EEG Dynamics Permit a Pre-diction of Epileptic Seizures?

Epilepsy patients suffer from so far unforeseeable seizures.To enable short term medical intervention a reliable seizureprediction method is desired. Recently, several seizureprediction methods have been proposed. We present amethodology to assess and compare seizure predictionmethods. It is applied to statistically evaluate severalprediction methods. Furthermore, we discuss electroen-cephalogram related characteristics of seizure predictionmethods. Based on these, extensions of existing predictionmethods are suggested.

Matthias WinterhalderCenter for Data Analysis and ModellingUniversity of Freiburg, [email protected]

Bjoern Schelter, Thomas Maiwald, Ariane SchadCenter for Data Analysis and ModelingUniversity of Freiburg, [email protected],

DS05 Abstracts 179


Armin Brandt, Andreas Schulze-BonhageEpilepsy CenterUniversity Hospital Freiburg, [email protected],[email protected]

Jens TimmerUniversity of FreiburgDepartment of [email protected]

MS100

The Stability of Periodic Patterns in Singular Per-turbed Reaction-Diffusion Equations

Recently, the stability of localized pulse solutions to sin-gular perturbed reaction-diffusion equations of Gray-Scottand Gierer-Meinhardt type has been established with theuse of Evans function methods. Apart from these localizedsolutions, these equations also posses large classes of oftenhighly non-trivial, stationary, spatially periodic patterns.In the present talk, the stability of these solutions will bestudied with the use of a generalization of the Evans func-tion based on Floquet theory. In the context of the gen-eralized Gierer-Meinhardt equation, it will be shown thatthe spectra associated to the stability of the patterns canbe computed explicitly. Moreover, bifurcations betweendifferent patterns, and the associated changes in stability,can be studied in full detail. This talk is based on jointwork with Harmen van der Ploeg (Amsterdam).


MS100

Mixing in the Kuppers-Lortz Mode

Keith JulienUniversity of [email protected]

MS100

New Results on the Stability of Pulse-like Solutionsto a Coupled Nonlinear Klein-Gordon System

When subject to sufficient twist, an elastic filament keptunder tension typically undergoes a writhing bifurcation.Near threshold, the corresponding dynamics may be mod-eled by two coupled nonlinear Klein-Gordon equations,which are envelope equations for the amplitudes of the lo-cal deformation and twist of the filament. I will presenta necessary and sufficient condition for the spectral sta-bility of pulse solutions to these envelope equations. Thisresult, obtained by Hamiltonian methods, completes andextends the analysis of S. Lafortune and J.L. (Physica D182, 103-124 (2003)), in which a sufficient condition for theinstability of “non-rotating” pulses was found by means ofEvans function techniques. I will also discuss a differentway of numerically evaluating the Evans function, and testit against the above analytical results. This work is jointwith Stephane Lafortune and Silvia Madrid-Jaramillo.

Joceline LegaUniversity of Arizona, USA

[email protected]

MS100

Localized Structures in Two-dimensional PhotonicLattices

Ziad MusslimaniUniversity of Central FloridaDepartment of [email protected]

MS101

Extracting Low-Order Stochastic Models FromData

We present a numerical technique to derive optimalstochastic models (Markov chain or SDEs) for the descrip-tion of the evolution of a few interesting collective variablesin large-sized dynamical systems. The technique is basedon constructing the stochastic model whose eigenfunctionsand eigenvalues are the closest (in some appropriate norm)to the one gathered from the observations. The techniqueis validated on an example arising from climate dynamics,namely by extracting a stochastic model for the evolutionof the first few principal orthogonal modes observed in ageneral circulation model.


Daan CrommelinCourant InstituteNew York [email protected]

MS101

Mathematical and Computational Strategies forStochastic Multiscale Problems: Coarse-Graining,Loss of Information

Markos A. KatsoulakisUMass, AmherstDept of [email protected]

MS101

Analysis of Multiscale Methods for Stochastic Dif-ferential Equations

We analyze a class of numerical schemes proposed in [25]for stochastic di erential equations with multiple time-scales. Both advective and di usive time-scales are con-sidered. Weak as well as strong convergence theorems areproved. Most of our results are optimal. They in turn al-low us to provide a thorough discussion on the e ciency aswell as optimal strategy for the method.

Di LiuCourant Institute, New York [email protected]

MS101

Modulation Equations: Stochastic Bifurcation in

180 DS05 Abstracts

Large Domains

We consider the stochastic Swift-Hohenberg equation on alarge domain near its change of stability. We show that,under the appropriate scaling, its solutions can be approxi-mated by a periodic wave, which is modulated by the solu-tions to a stochastic Ginzburg-Landau equation. We thenproceed to show that this approximation also extends tothe invariant measures of these equations. This is jointwork with D. Blomker and M. Hairer.

Grigorios PavliotisImperial [email protected]

MS101

Stochastic Modelling of Complex Dynamics Ex-hibiting Metastability


MS101

Reduced Dynamics for a Class of Conservative Sys-tems

Novel approach to derivation of reduced dynamics for aclass of energy-conserving systems is discussed. Asymp-totic analysis and microcanonical averaging are utilized toderive closed-form reduced equations for long-term evolu-tion of slow subset. No ad-hoc assumptions for the sta-tistical properties of the fast heat bath are made; instead,single microcanonical simulation is utilized to estimate sta-tistical properties of the fast modes for all energy levels.Truncated Burgers-Hopf model is utilized as an example.

Andy [email protected]


Ilya TimofeyevDept. of MathematicsUniv. of [email protected]

MS102

Torus Destruction and Chaotic Behavior in a Sys-tem of Two Coupled Oscillators

A one mass system with two degrees of freedom is con-sidered here. Using the normal form method of averagingwe derive conditions for stability of the periodic solutionand spot a T2-torus by a Neimark-Sacker bifurcation. Weshow how this torus breaks down, through one period dou-bling (Shilnikov’s scenario), by numerically following thechanges in the involved manifolds. The strange attractorand chaotic behavior following the breakdown of the torusare studied in detail.

Taoufik BakriMathematics Institute, Utrecht University,PO Box 80.010 3508 TA Utrecht

[email protected]

MS102

Quasi-Periodicity in Dissipative and ConservativeSystems

Kolmogorov-Arnold-Moser (of KAM) theory was devel-oped for conservative dynamical systems that are nearlyintegrable. Integrable systems in their phase space usuallycontain lots of invariant tori, and KAM theory establishespersistence results for such tori, which carry quasi-periodicmotions. We sketch this theory, which begins with Kol-mogorov’s pioneering work (1954).

Henk BroerUniversity of GroningenDepartment of [email protected]

MS102

Hamiltonian Torus Bifurcations Related to SimpleSingularities

Heinz HanssmanTechnische Universiteit [email protected]

MS102

Computation of Invariant Tori in Quasi-PeriodicSystems

We explain the parameterization method to compute in-variant tori and their whiskers in quasi-periodic systems.We apply the method to study two examples, that arequasi-periodic perturbations of the Hnon map and thestandard map.


Alex HaroUniversitat de BarcelonaUniversitat de Barcelona Gran Via 58508007 Barcelona(Spain)716865

MS102

Interactions of Torus Bifurcations in Maps with Di-mension ¿2

In this talk I describe two remaining cases of codimension2 bifurcations for maps. First a normal form analysis isgiven, which in turn is applied to an example. These bi-furcations need maps with dimension at least 3 or 4, butmechanical systems with only a few building blocks maydisplay such bifurcations already as we will show.

Hil MeijerUniversity of UtrechtMathematics [email protected]

DS05 Abstracts 181

MS102

Continuation ofQuasi-Periodic Tori

We present a new algorithm for the continuation of quasi-periodic invariant tori of ordinary differential equations insufficiently many parameters. The proposed method re-quires neither the computation of a base of a transversalbundle, nor re-meshing during continuation. It is inde-pendent of the stability type of the torus and can step overquasi-periodic bifurcation points. We demonstrate the per-formance of the method with examples.

Frank SchilderDepartment of Engineering MathematicsUniversity of [email protected]

MS103

Routes to Pull-in

We study the nonlinear dynamics of electrostatic MEMS todetermine the underlying mechanisms leading to the pull-inphenomenon. Pull-in occurs due to the non-existence of anequilibrium configuration corresponding to the applied DCvoltage. We found several dynamic mechanisms where pull-in developed even though there was a stable equilibriumposition corresponding to the applied DC voltage. Any ofthese mechanisms can lead to pull-in, depending on theexcitation level and initial conditions of the device.

Eihab Abdelrahman, Ali NayfehVirginia TechDepartment of Engineering Science and [email protected], [email protected]

MS103

Discontinuity-Driven Design and Control of ImpactMicroactuators

Impact microactuators rely on repeated collisions to gener-ate gross displacements of machine elements. Their designrelies on an understanding of the critical transition betweennonimpacting and impacting dynamics. A normal-formanalysis is presented that predicts the character of suchtransitions from a set of conditions that are computable interms of system properties at grazing. The analysis alsosuggests opportunities for using passive design or activecontrol to regulate the near-grazing response.


Xiaopeng ZhaoDepartment of Engineering Science and MechanicsVirginia Polytechnic Institute and State [email protected]

MS103

Effects of Squeeze Film Damping on Stability andDynamics of Microbeam-based MEMS

Damping in microelectromechanical systems strongly af-fects their performance, design, and control. Typical mi-croresonators employ a parallel-plate capacitor, in whichone plate is actuated electrically and its motion is detectedby capacitive changes. Under such conditions a squeeze-

film damping appears. This phenomenon introduces a fre-quency (and consequently an amplitude) dependent force.In this research the contribution of the squeeze-film phe-nomenon in MEMS dynamics is simulated by a nonlinearamplitude dependent function.

G. N. JazarNorth Dakota State [email protected]

MS103

Touchdown Dynamics in Electrostatic MEMS

John A. PeleskoUniversity of [email protected]

MS103

Flow Control inside Micro-Fluidic Systems: Mod-eling, Sensing, and Feedback Control Design

Micro-fluidic technology has the potential to allow hand-held devices with the functionality of existing biologicaland chemical laboratories, it can be used to create wearableor implantable drug delivery platforms, and it allows directhandling of biological materials such as cells, proteins, andDNA. In our research we have found that feedback controlis often required for robust micro-fluidic performance (asin the UCLA electro-wetting devices) and it allows newcapabilities in other cases (as in our multi-particle steer-ing system). This talk will address our efforts to integrateresearch in system design and feedback control with therapid progress being made in micro-fluidic systems. Re-sults will be shown for two physical systems. The first isthe Electro-Wetting-On-Dielectric (EWOD) system devel-oped at UCLA by CJ Kim. Here a grid of electrodes is usedto locally change surface tension forces on liquid droplets:by choosing the electrode firing sequence it is possible tomove, split, join, and mix liquids in the droplets. I willdescribe our modeling, vision sensing, and control resultsfor this system. In particular, I will show our algorithmsfor precision control of droplet splitting and joining, andI will describe how feedback can be used to correct foran unknown external environment. The second system is amicro-fluidic ”no-laser tweezer” system that can be used tosteer many particles at once. I will show how we use flowcontrol to create an underlying, time-varying fluid flow thatcarries all the particles at once along their desired trajec-tories, and I will describe the status of our experiments atMaryland and at NIST. Because the system does not re-quire lasers and high quality optics with long optical pathlengths, it is cheap and it can be miniaturized. This sys-tem is being used to steer cells for a ”cell clinics” project atthe University of Maryland. I will close the talk by outlin-ing some of the key motivations, benefits, bottlenecks, andremaining open challenges in integrating feedback controland micro-fluidic systems research.

Benjamin ShapiroUniversity of MarylandCollege Park, [email protected]

MS103

Generalized Parametric Resonance in Electrostat-

182 DS05 Abstracts

ically Driven MEM Oscillators

Steven W. ShawDepartment of Mechanical EngineeringMichigan State [email protected]

Jeffrey RhoadsDept. of Mechanical EngineeringMichigan State [email protected]

Jeffrey Moehlis, Barry DemartiniDept. of Mechanical EngineeringUniversity of California-Santa [email protected], [email protected]

Kimberly TurnerDept Mech Env EngUC Santa [email protected]

MS104

Shanon’sLimit for a Class of Nonlinear Circuits

We will consider a class of singularly perturbed nonlineardifferential equation under external forcing via noisy chan-nels modeling a modulation scheme of converting analogsignals to digits. Shannon’s information limit is defined forthis class of equations and is shown to be computable byusing a PDE method based on the Fokker-Planck equation.Generalizations of this method will also be given.

Shui-Nee ChowSchool of Mathematics,Georgia Institute of Technology, Atlanta, GA [email protected]

MS104

Dynamics of Infinite-dimensional Stochastic Sys-tems

We present a non-linear multiplicative ergodic theoremfor infinite-dimensional stochastic semiflows on a Hilbertspace. The theorem yields the existence of local stable andunstable manifolds in the neighborhood of hyperbolic sta-tionary solutions of semilinear stochastic evolution equa-tions and stochastic functional differential equations.

Salah MohammedDepartment of MathematicsSouthern Illinois University, Carbondale, IL [email protected]

MS104

Principal Spectrum Theory for Random ParabolicEquations

In the current talk, I will present some recent joint workwith Janusz Mierczynski on principle spectrum for randomparabolic equations. Among others, I will show that theprincipal Lyapunov exponent of a random parabolic equa-tion is greater than or equal to the principal eigenvalue ofthe associated time-averaged equation, which has an im-portant biological implication, that is, heterogeneous envi-ronment favors population’s persistence. I will also discuss

some application to competition reaction diffusion systems.

Wenxian ShenDepartment of MathematicsAuburn University, Auburn, AL [email protected]

MS104

The Effects of Noise on Transient Pattern Forma-tion

Transient phase separation processes can create compli-cated evolving patterns. Typical underlying models ofteninclude both deterministic and stochastic partial differen-tial equations, and it is therefore natural to ask in whatway stochasticity impacts the dynamics and the geometryof the produced patterns. In this talk I will present boththeoretical and computational results addressing similari-ties and differences in stochastic models which are obtainedfrom deterministic ones by adding an additive noise termof variable intensity.


MS105

Periodic Orbits and Chaotic Sets in a Low-Dimensional Model for Turbulent Shear Flows

We analyse a low-dimensional model for turbulent shearflows. The model is derived for sinusoidal shear flow, inwhich fluid between two free-slip walls experiences a sinu-soidal body force, and is based on Fourier modes represent-ing important physical structures. The model illustratesmany phenomena observed and speculated to exist in thetransition to turbulence, including subcritical and inter-mittent transition, exponential distributions of turbulentlifetimes, and unstable equilibria and periodic orbits.

Holger FaisstPhilipps Universitat MarburgFachbereich [email protected]



MS105

POD-Galerkin Models of Open Shear Flows: WhatCan we Expect of Them?

The last two decades of theoretical fluid mechanics haveseen increased interest in low-dimensional models of flows.However, the reduced complexity of low-dimensional mod-els does comes at a cost that is not always well understood.We will look at the way low-dimensional models can com-plement the toolkit of the fluid dynamicist, and what theirlimitations are. The use of low-dimensional models leads

DS05 Abstracts 183

to questions that are sometimes ignored, at a peril.

Dietmar RempferIllinois Institute of [email protected]

MS105

Control-Oriented Models of Channel Flow

Reduced-order models of linearized flow in a plane chan-nel are presented, using several different model-reductiontechniques, including Proper Orthogonal Decomposition(POD), balanced truncation, and a new method called bal-anced POD. These models are used to obtain observers andcontrollers to damp disturbances, and to explore funda-mental limits in performance. The eventual goal is to usefeedback to stabilize the laminar flow and delay transitionto turbulence.

Clancy W. RowleyPrinceton [email protected]

MS105

Stability and Accuracy of Periodic Flow SolutionsObtained by a POD-Penalty Method

We develop a new penalty method to derive low-dimensional Galerkin models for fluid flows with time-dependent boundary conditions. We then outline a proce-dure based on bifurcation analysis in selecting the propervalues of the penalty parameter(s) that yield asymptoti-cally stable periodic solutions of the highest possible ac-curacy. We illustrate this new approach by studying flowpast a circular cylinder using direct numerical simulationdata, and a wave-structure interaction problem using par-ticle image velocimetry data.

George KarniadakisBrown [email protected]

Sirod SirisupBrown UniversityDivision of Applied [email protected]

MS105

Coarse Dynamics of Plane Couette Flow

We describe a technique for the efficient computationof plane Couette flow when only a well-resolved andcomputationally-costly simulation is available. Instead ofusing the standard POD-Galerkin technique, we adopt aframework which does not require explicit equations for theevolution of these coherent structures. Rather, a computa-tional superstructure is designed to combine short burts ofDNS with relatively coarse integration of the large scalesto extract the dynamics of the dominant features.

Troy R. SmithCalifornia Institute of [email protected]

MS105

Control Oriented Empirical Galerkin Models and

Model Based Design for the Cylinder Wake

Control oriented Galerkin flow models must combine sim-plicity and ample dynamic range. Enablers include ‘sub-grid’ estimation of turbulence and pressure representations,modes from multiple operating points, and actuation mod-els. An invariant manifold defines the models dynamicenvelope. It must be respected and can be exploited inobserver and control design. These ideas are benchmarkedin the cylinder wake system and validated by a systematicDNS investigation of a 4-dimensional Galerkin model ofthe controlled wake.

Bernd NoackTechnical University [email protected]

Gilead TadmorNortheastern [email protected]

MS106

The Colley Matrix Ranking Method

The National Championship in college football is, to myknowledge, unique in major sports in that it features aplayoff of only two teams in a single game to determine thenational champion. While accurate seeding is obviously im-portant in any tournament, nowhere is it more paramountthan in a two-team, one-game playoff. As such, severaldifferent inputs are used, including ”human” polls, whichamount to surveys of sportswriters and coaches across thecountry, but also six computer ranking systems, of whichthe Colley Matrix is one. The method was developed withsimplicity in mind, focusing only upon wins and losses,and ignoring such factors as margin of victory, chronologyand play at home or away. The method is, as such, less apredictor of future outcomes than a hindcaster of accumu-lated merit, and is therefore ideal for determining whichteams ”deserve” to play for the national championship.The mathematical method is an iterative extension of asimple formula used by Laplace to determine probabilitiesof binary outcomes (such as football games). The iterativemethod has an available linear solution, called the ColleyMatrix method.

Wes [email protected]

MS106

Adaptive Scheduling

Conventional schedules have been found to be insufficientin many sports to determine the best k teams (Gibbons,Olkin, and Sobel, 1978; Weiss, 1986). Tournaments, bybeing adaptive, determine the best teams with greater pre-cision and efficiency. However, the methods used to seedteams in tournaments is suboptimal. We suggest an adap-tive scheduling algorithm that attempts to optimize theprobability of correctly selecting the best k teams, anddemonstrate the method with college football data.

Mark JohnsonSt. Louis [email protected]

Matthew S. Johnson

184 DS05 Abstracts

Baruch College, City UniversitySportMetrikamatthew [email protected]

MS106

A Multi-Dimensional Ranking Approach to Intran-sitive Relations

Modern information systems are challenged with extract-ing useful information from large data sets collected fromsprawling, and potentially unreliable networks. A strik-ingly similar task is to accurately rate football teams, whichplay short disparate schedules, and whose performancemay vary significantly from week to week. Many modelsprovide a one-dimensional ’rating’ of a team’s strength. Wewill discuss the limits of such schemes and explore general-ized approaches to multi-dimensional evaluations for teamsin competitive sports.

Ken MasseyHollins UniversityDept of [email protected]

MS106

Random Walker Rankings for College Football

Using only win-loss information of games, we develop afamily of rankings defined by random walks on the bi-ased graph of teams (vertices) and games played (edges).These ranking rules are easily explained in terms wherethe random walkers represent fickle voters. We investigatestatistical properties and examine the asymptotics of therankings at extreme values of the bias parameter. We alsoinvestigate the connection between the rankings and theunderlying community structure of the network of gamesplayed.

Mason A. PorterGeorgia Institute of TechnologySchool of Mathematics and Center for Nonlinear [email protected]

Peter J. MuchaGeorgia Institute of TechnologySchool of [email protected]

Thomas CallaghanSchool of MathematicsGeorgia Institute of [email protected]

MS106

Probabilistic Models for Tennis

A model for computing the probability of winning a game,a set, and a match in tennis is described, based on eachplayer’s probability of winning a point on serve. Both twoout of three and three out of five set matches are consid-ered, allowing a 13-point tiebreaker in each set, if necessary.The question of whether it is advantageous to serve firstor receive first is answered. Then the probability of eachplayer winning a 128 player tournament is calculated. Datafrom the 2002 US Open and Wimbledon tournaments areused both to validate the theory as well as to show how pre-dictions can be made regarding the ultimate tournament

champion.


Kamran [email protected]

Joseph KellerStanford [email protected]

MS106

Least Squares-Gaussian Sports Predictions: SomeFundamental Conclusions

Since 1971, I have been using a least squares approachto rate teams in American college and professional foot-ball, American college basketball, Australian Rules foot-ball, European soccer (England, Italy, Norway and Ger-many), Super 12 rugby union and the Zurich Premiershipin rugby union. The ratings are based on win margin ad-justed for home advantage. Predictions follow using theratings, a shrinking factor, and home advantage. The re-sults of those 30+ years provide insight into sports andsports predictions, such as the relative accuracy of variousprediction schemes (no real difference, actually), home ad-vantage in various sports (regular season and playoff com-petition), ease of scoring as it affects predictability andgambling strategies.

Ray StefaniCal State Long [email protected]

MS107

Coherent Spontaneous On-going Activity in Cortex

It has been shown experimentally that spontaneous cor-tical activity in the absence of sensory inputs modulatesstimulus evoked activities and is correlated with behavior.In the visual cortex, there is a close relationship betweenongoing spontaneous activity and the spontaneous firing ofa single neuron. There are dynamical switching amongstthese spontaneous cortical states, which may span sveralhypercolumns spatilaly and are closely associated to ori-entation maps. We will present our theoretical modelingresults to illustrate a possible mechanism underlying thisspontaneous cortical activity and discuss further experi-mental evidence consistent with our theoretical model.

David CaiCourant instituteNew York [email protected]

MS107

Experimental Status of Electical Waves In Vivo

David KleinfeldPhysicsUniversity of California at San [email protected]

DS05 Abstracts 185

Samar MehtaNeurobiologyUviversity of California at San [email protected]

MS107

Reverse-Correlation Techniques and Cortical Ar-chitecture

Reverse-time correlation measurments give the average ori-entation dynamics of individual neurons within a highlyexcited visual cortical neuronal network. The resulting ori-entation tuning curves provide specific information aboutthe nature of cortico-cortical connections, in particular, thestrength and extent of cortical inhibition. We present a setof models that uncover and explain the connection betweenthe experimentally observed tuning curves and the relevantcortical architecture.

Gregor KovacicRensselaer Polytechnic InstDept of Mathematical [email protected]

MS107

Kinetic Theories of Neuronal Networks

We will compare various closures in the reduction ofintegrate-and-fire neuronal networks to kinetic theory. Wewill discuss mathematical aspects of these kinetic theoriesand present results of the kinetic theories and full numeri-cal simulations of integrate-and-fire neuronal networks.

Adi RanganCourant Institute of Mathematical SciencesNew York [email protected]

MS107

Synchrony-Dependent Propagation of Signals inNetworks of Neurons

To examine experimentally how neural signals are trans-mitted, a feedforward network of live neurons was repro-duced in brain slices using a computer-driven iterative pro-cedure. When inputs were delivered to the first layer,neurons in successive layers fired synchronously. Althoughsynchrony was robust and persisted under a wide range ofnetwork configurations, its spread can be limited by lateralinhibition. Synchrony appears to be an important meansof encoding and transmitting signals through neural net-works.

Alex Reyes, Cristina MarchettiCenter for Neural ScienceNew York [email protected], [email protected]

MS107

Panel Discussion

We have brought together neurophysiologists, modelersand theoreticians to highlight the recent work on wavesand coherent structures of neural activity. We end thistwo-part minisymposium on ”Waves and Coherent Struc-tures in Neural Systems” with a panel discussion.

Louis Tao

Department of Mathematical SciencesNew Jersey Institute of [email protected]

MS108

Modulated Structures in a Modulationaly StableOptical Interferometer

Evidence of periodic dissipative structures with an intrin-sic wavelength in a nonlinear optical system devoid of Tur-ing instability is given. They are found in the transversefield distribution of a nonlinear interferometer formed by aLiquid Crystal Light Valve with feedback, operating nearnascent bistability. Their existence is related to a ro-bust transition from flat tomodulated rotationally invari-ant two-dimensional fronts. The analytical expression ofthe threshold associated with that transition as well as thewavelength of the emerging structure are derived. Thesepredictions are in close agreement with numerical simula-tions.

Gregory P. KozyreffOCIAM, Mathematical Institute, Oxford [email protected]

MS108

Effects of Transverse Flow on Pattern Formation ina Nematic Liquid Crystal Layer with Optical Feed-back : Experiments and Theory

In a previous study we have evidenced that the presence ofa transverse flow in a one dimensional (1D) optical systemleads to convective instability patterns. Here, we derive thedifferent types of convective 2D patterns and their convec-tive and absolute thresholds. We show e.g. that one typeof patterns is always convective, and that the presence oftransverse flow can lower the threshold of pattern forma-tion. Experiments and theory are in very good agreement.

Eric LouvergneauxLaboratoire de Physique des Lasers Atomes et MoleculesUniversite des Sciences et Technologies de [email protected]

Christophe SzwajUniversite de Lille (France)Laboratoire [email protected]

Pierre GlorieuxLaboratoire PHLAM, CERLAUniversite de Lille 1 (FRANCE)[email protected]

Gonzague AgezPhLAM - Universite de Lille (France)[email protected]

Majid TakiUniversite des sciences et technologies de [email protected]

MS108

Nonlocality and Convective Instability in DiffusiveSystems.

We consider a large class of diffusive systems with nonlocalnonlinearity. We show that the stability of all the uniform

186 DS05 Abstracts

states of this class of systems is determined by a singledispersion relation with two parameters. By using a non-perturbative analytical approach we are able to analysethis dispersion relation and show that there are cases inwhich the convective instability window is so large that inpractise only noise sustained patterns can be observed.

Francesco PapoffDepartment of Physics, University of Strathclyde,[email protected]

Roberta ZambriniDepartment of PhysicsUniversity of [email protected]

MS108

Bistability Between Different Dissipative Solitonsin Nonlinear Optics

We report the observation of different localized structurescoexisting for the same parameter values in an extendedsystem. The experimental findings are carried out in anonlinear optical interferometer, and are fully confirmedby numerical simulations. The existence of each kind oflocalized structure is put in relation to a correspondingdelocalized pattern observed. Quantitative evaluation ofthe range of pump parameter allowing bistability betweenlocalized structures is given. The phenomenon reportedresults to be robust in parameter space.

Gregory KozyreffMathematical InstituteOxford [email protected]

Mustapha Tlidiuniversite Libre de BruxellesOptique nonlieaire [email protected]

Umberto BortolozzoIstituto Nazionale di Ottica ApplicataFirenze, [email protected]

Pier Luigi RamazzaIstituto Nazionale di Ottica ApplicataFlorence, [email protected]

Luc PasturUniversity of OrsayParis, [email protected]

MS108

Pattern Formation in a Liquid Crystal Light ValveThrough Alternation of Dynamics

It has recently been discovered that modulation of param-eters in spatially extended nonlinear systems can have astriking impact on pattern formation. For example, rapidlyswitching between two states, each of which is homoge-neous, can lead to the formation of patterns. We havefound experimentally similar phenomena, not previouslydescribed, in a spatially extended nonlinear optical sys-

tem (LCLV with feedback). Our presentation describesthe experimental system and the conditions under whichpatterning occurs.

John SharpeCalifornia Polytechnic State University,[email protected]

Pier Luigi RamazzaIstituto Nazionale di Ottica ApplicataFlorence, [email protected]

Nilgun Sungar, Karl SaundersCal [email protected], [email protected]

MS108

Nonlinear 2D Diffractive Feedback Systems: A Callfor Applications

In the presentation we consider such applications of thenonlinear 2D-feedback systems as parallel image processingand atmospheric and adaptive optics.

Mikhail VorontsovThe Army Research Laboratory and the University ofMaryland,College Park, [email protected]

MS109

Bifurcation of Relaxation Ocillations

The talk deals with bifurcation of relaxation oscillations intwo dimensions, emphasizing the transient canard oscilla-tions. It relies on joint work with Robert Roussarie.

Freddy DumortierLimburgs Universitair Centrum, [email protected]

MS109

Periodic Waves in a Class of Singularly PerturbedDiffusive Two-Predator-One-Prey Systems

We consider a class of singularly perturbed diffusive two-predator-one-prey systems and establish the existence ofperiodic traveling waves which provide mechanism for co-existence. The main tool employed is the geometric singu-lar perturbations for turning points and invariant manifoldtheory.

Weishi LiuUniversity of [email protected]

MS109

Asymptotic Expansions for the Lagerstrom Model:A Geometric Approach

The present work is a continuation of our geometric sin-gular perturbation analysis of the Lagerstrom model prob-lem. We reinterpret Lagerstrom’s equation as a dynamicalsystem which we analyze by means of invariant manifoldtheory as well as of the blow-up technique. We derive rig-

DS05 Abstracts 187

orous asymptotic expansions for the Lagerstrom problemwithin this framework, thereby establishing a connection tothe method of matched asymptotic expansions. We explainthe structure of these expansions and demonstrate that theoccurrence of the well-known logarithmic switchback termstherein is caused by a resonance phenomenon.

Nikola PopovicBoston UniversityCenter for BioDynamics and Department of [email protected]

MS109

Combustion Fronts in Porous Media With TwoLayers

We investigate a simplified model of combustion in a porousmedium with two layers. Each layer admits a travelingcombustion wave with a certain speed, and heat can diffusebetween the layers. We show: (1) If the coefficient of heatdiffusion between the layers is small, and the speeds of thetwo traveling waves are close, then the two-layer systemadmits a traveling combustion wave. (2) If the coefficient ofheat diffusion between the layers is large, then the two-layersystem admits a traveling combustion wave, regardless ofwhether the speeds of the individual traveling waves areclose. This wave is approximately an average of waves inthe individual layers. The proofs use geometric singularperturbation theory. Numerical simulations indicate thatthese traveling waves are the dominant feature of solutions.

Jesus Carlos da MotaInstituto de Matematica e EstatisticaUniversidade Federal de Goias (Brazil)[email protected]

Stephen SchecterNorth Carolina State UnivDepartment of [email protected]

MS109

Blow-Up Analysis of Delayed Hopf-Bifurcations

Differential equations with slowly varying parameters canbehave quite differently from the corresponding static bi-furcation problems. In the case of Hopf bifurcations thewell known phenomenon of bifurcation delay occurs in an-alytic systems. We give a geometric analysis of this phe-nomenon. The analysis is based on the choice of a suitableintegration path in complex time and the blow-up methodfor singularly perturbed differential equations.

Peter SzmolyanInstitute for Analysis and Scientific ComputingVienna University of [email protected]

MS109

Wave Equations with Strong Potentials

In this talk, we consider vector valued wave equationswhere there is a strong constraining potential. This poten-tial achieves its minimum on a submanifold. We study thelimit of solutions of finite energy as the constraining poten-tial tends to infinity. This problem can also be viewed asrealizing holonomic constraints (to submanifolds in config-uration spaces) to Hamiltonian motion by using a strong

constraining potential. We rigorously justify the formallimit obtained by multi-scale expansion.

Chongchun ZengUniversity of VirginiaDepartment of [email protected]

MS110

High-Dimensional Chaos in the Kuramoto Model

A novel high-dimensional chaotic behavior is found in theKuramoto model of coupled phase oscillators. A half ofthe Lyapunov spectrum appears to be positive and theLyapunov dimension almost attains the total system di-mension. We find that the phase chaos phenomenon istypical for different distributions of the natural frequenciesand also, for other ensembles of oscillators both regular andchaotic, e.g. for networks of coupled Rssler systems.

Yuri L. MaistrenkoInstitute of Mathematics, Kiev, UkraineForschugszentrum Juelich, [email protected]

MS110

Synchronization, Desynchronization and Noise inElectroreceptors of paddlefish

The electrosensory system of paddlefish is used as a modelsystem to study general mechanisms of synchronizationand desynchronization. Our approach is to create neuro-electronic models, composed of afferents from two differentelectroreceptors in an in vivo preparation of the paddle-fish, linked via a computer interface. Interface providesfeedback to drive afferents to bursting regimes and cou-pling between electroreceptors. Using our hybrid model,we validated experimentally a new approach for disrupt-ing the synchronization of different oscillators, based onintermittent multisite transient phase resetting.

Alexander NeimanDepartment of Physics and AstronomyOhio [email protected]

David F RussellCenter for Neurodynamics, Univ. of Missouri at St. LouisSt. Louis, MO 63121, [email protected]


MS110

Complete Synchronization, Directed Percolationand Finite Amplitude Lyuapunov Exponents

Transition to complete synchronization in spatially ex-tended systems is analysed showing the existence of twoscenarios Besides the standard analogy with Kardar-Parisi-Zhang equation a connection with directed percolation isfound which is then investigated by implementing finite-amplitude Lyapunov exponents.

Roberto Livi

188 DS05 Abstracts

Dipartimento di FisicaFirenze, [email protected]

Antonio PolitiNational institute of Optics, Florence, [email protected]

Francesco GinelliDepartment of PhysicsWurzburg, [email protected]

Alessandro TorciniIstituto Sistemi Complessi CNRINOA - FIrenze, [email protected]

MS110

Effective Desynchronization of Coupled OscillatorsBy Nonlinear Delayed Feedback

We propose a new method for desynchronization of ensem-bles of interacting and strongly synchronized oscillators.We show that the stimulation input in the form of delayedself-feedback combined nonlinearly with instantaneous sig-nal can have a desynchronizing effect on the oscillations.The proposed method represents a robust and demand-controlled noninvasive technique with which different de-grees of desynchronization can be achieved and controlled.This makes the method particularly attractive for applica-tions, in particular, for deep brain stimulation.

Oleksandr PopovychInstitute of MedicineForschungszentrum Juelich, [email protected]

MS110

Rotating Spiral Waves with Phase-RandomizedCore in Nonlocally Coupled Oscillators

Yoshiki KuramotoDepartment of MathematicsHokkaido University, Sapporo , [email protected]

Shin-ichiro ShimaDepartment of Physics,Kyoto University, Japans [email protected]

MS110

Model-Based Development of DesynchronizingDeep Brain Stimulation

For the therapy of severe neurological diseases permanenthigh-frequency (¿ 133 Hz) deep brain stimulation (DBS)is performed. This treatment has been developed heuris-tically. To improve treatment with DBS, with methodsfrom nonlinear dynamics and statistical physics novel DBStechniques were designed to restore the physiological firingmode by mild but effective desynchronization. In a pilotstudy in patients during depth electrode implantation thenovel DBS technique turned out to be superior.

Peter A. Tass

Institute of Medicine (MEG)Research Centre [email protected]

PP0

Lyapunov Exponent Analysis of Biological Signal-ing Network Dynamics

We employ a finite-time Lyapunov exponent analysis ap-proach to identify separatrices in phase space, delineat-ing qualitatively diverse signaling network dynamics gov-erning pre-steady-state cell fate decisions. Application ofthis approach to a model of the protein network regulatingcell death responses to cytokines yields insights concerningthis regulation not obtainable using steady-state bifurca-tion analysis. This approach should facilitate similar in-vestigations of larger signaling networks.


Peter SorgerDepartment of Biology, Division of Biological [email protected]

Douglas LauffenburgerDivision of Biological Engineering, Dept. of Chem. [email protected]

Bree B. AldridgeMassachusetts Institute of TechnologyBiological Engineering [email protected]

PP0

Noise-Induced Dynamics in a Free-Electron Laser

We study the effect of noise on the dynamics of free-electron laser oscillators. We find that the small amount ofnoise present in any experiment leads to a behavior quali-tatively different from the noise-free (deterministic) equiv-alent. This is attributed to the nonnormality of the modesinvolved, induced by the convective nature of the system.The results are found very similar to the ones obtained inthe case of mode-locked lasers.

Serge Bielawski

PhLAM/Universite Lille I,[email protected]

Christelle Bruni, Gian-Luca Orlandi, M.-E. CouprieCEA/[email protected],[email protected], [email protected]

CHRISTOPHE SwajPhLAM/Universite Lille [email protected]

DAVID GarzellaCEA/LURE

DS05 Abstracts 189

[email protected]

PP0

Dynamical Models of Finger Biomechanics andNeuromuscular Control

We analyze the periodic motor patterns of middle fingercontrol of a trackball, during a psychophysics task in whichthe subject has to match a constant velocity using thetrackball. We explore minimal hybrid (DAE) kinematicmodels of the system, and investigate the roles of feedbackcontrol both from ‘pre-flexes’ and from central motor sys-tems.


Robert ClewleyDepartment of MathematicsCornell [email protected]

Francisco Valero-CuevasNeuromuscular Biomechanics LaboratoryCornell [email protected]

PP0

Pattern Selection in a Hopf Bifurcation Forced atMultiple Resonant Frequencies

In Faraday surface waves complex super-lattice patternsand quasi-patterns can be excited by multi-frequency forc-ing. Motivated by this observation we investigate the com-petition between various spatially periodic patterns in acomplex Ginzburg-Landau equation describing a Hopf bi-furcation with frequency ω forced simultaneously with fre-quencies ω, 2ω, and 3ω. From the Ginzburg-Landau equa-tion we derive amplitude equations for various planformsand study their stability analytically. We then investi-gate their dynamics in simulations of the Ginzburg-Landauequation.


Jessica ConwayNorthwestern UniversityApplied [email protected]

PP0

MatCont: a Software Package for Dynamical Sys-tems with Applications to Modelling Neural Activ-ity.

MatCont is a graphical MATLAB package for the inter-active numerical study of a range of parameterized non-linear problems. The current version is freely availableat: http://allserv.rug.ac.be/ ajdhooge where also a slightlymore general non - GUI version Cl MatCont is available.Both MatCont and Cl MatCont allow to compute curvesof equilibria, limit points, Hopf points,limit cycles and flip,fold, torus and branch points of limit cycles. It does branch

switching at branch points of equilibria and limit cyclesand detects limit points, Hopf points, limit points of cy-cles, branch points of cycles ... MatCont makes the MAT-LAB odesuite for time integration interactively availableand can use the MATLAB Symbolic Toolbox for comput-ing derivatives whenever it is installed. In the case of limitcycles MatCont discretizes the BVP exactly as in AUTOand CONTENT, i.e. by orthogonal collocation. The sys-tems that arise in this way are typically sparse and theirsparsity increases with the number of test intervals used inthe discretization. In MatCont and Cl MatCont the spar-sity of the linearized systems is exploited by using the Mat-lab sparse matrix routines. Among the recent additions toMatCont we mention the computation of normal form co-efficients for bifurcations of limit cycles. We present theuse of MatCont in a few interesting neural models.

Yuri A. KuznetsovMathematical InstituteUtrecht [email protected]

Annick DhoogeGhent UniversityDepartment of Applied Mathematics and [email protected]

Willy GovaertsDept. of Applied Mathematics and Computer ScienceGhent [email protected]

PP0

A Biophysical Model for Sleep-Wake Timing

The transitions between states of sleep and wake are regu-lated by a network of neurons in the brainstem and hy-pothalamus. We model this network as a collection ofcoupled oscillators, and we incorporate circadian, home-ostatic, and other biophysical elements to simulate bothnormal and pathological transitions. By exploiting geomet-ric properties of the network, we are also able to capturewaking behaviors on two timescales: seconds and minutes.


Cecilia Diniz BehnBoston UniversityDepartment of Mathematics and Center for [email protected]

Thomas Scammell, Takatoshi MochizukiDepartment of NeurologyBeth Israel Deaconess Medical [email protected],[email protected]

Emery BrownDepartment of Anesthesia and Critical CareMassachusetts General [email protected]

PP0

Bifurcations of Stable Sets in Noninvertible Planar

190 DS05 Abstracts

Maps

Many applications give rise to systems that can be de-scribed by noninvertible maps. These fold the phase space,so that different regions have different numbers of pre-images. This makes computing backward orbits and stablesets more complicated as they may cross critical curveswhere the number of pre-images changes. We compute thestable sets of a two-dimensional noninvertible map, whichdefine the boundaries of basins of attraction, using a re-cently developed algorithm.



James P. EnglandDepartment of Engineering MathematicsUniversity of [email protected]

PP0

Frequency Dynamics of Semiconductor Lasers Sub-ject to Filtered Optical Feedback

If part of the emission of a semiconductor laser is fed backinto the laser after passing through an optical filter, charac-teristic oscillations of the frequency on the time scale of thedelay time are found. By performing a bifurcation analy-sis of the corresponding delay differential equation modelwe show how such oscillations emerge from certain Hopfbifurcations of the basic solutions of the system, which areknown as external cavity modes.


Daan LenstraAfdeling Natuurkunde en SterrenkundeVrije Universiteit [email protected]

Hartmut ErzgraberVrije Universiteit AmsterdamAfdeling Natuurkunde en [email protected]

PP0

Reconnection/Collision Processes and Meanders inthe Standard Nontwist Map

Using the standard nontwist map as a single-harmonicmodel map for nontwist systems, meandering tori and pe-riodic orbits between dimerized odd-periodic orbit chainsare investigated. The presence of these ”non-KAM” orbitsgives rise to more intricate reconnection/collision sequencesthan previously observed. We present such non-standardsequences and numerically establish parameter ranges fortheir applicability. We further study the destruction ofthe ”central” torus, i.e., the meandering torus of extremal

winding number.

Alexander WurmDepartment of Physics, Fusion StudiesThe University of Texas at [email protected]


Kathrin Fuchss, P. J. MorrisonDepartment of Physics and Institute of Fusion StudiesThe University of Texas at [email protected], [email protected]

PP0

1,2,3 Some Examples of Local Bifurcations ofHigher Codimension

We present examples of local bifurcations of codimension1, 2 and 3 that appear in ecological models. The bifurca-tions are shown in three-parameter bifurcation diagrams.Among others we show an example of a codimension-31:1-resonant double Hopf bifurcation. This bifurcation isformed by codimension-1 Hopf bifurcation surface in theshape of a Whitney umbrella. The bifurcation diagramshave been produced by combining analytical calculationswith an algorithm for numerical triangulation of implicitlydefined surfaces. This process is also explained.


Dirk Stiefs, Ulrike FeudelICBM Theoretical Physics/Complex SystemsCarl von Ossietzky Universitaet [email protected], [email protected]

PP0

Phase Transitions in Equilibrium and Non-Equilibrium Systems with Xy-Model-Like Interac-tions

We study noise- or temperature-driven phase transitions ofvarious dynamical systems consisting of elements with XY-model-like interactions. For systems comprised of moving,self-propelled elements, a second-order phase transition isobtained. We establish an equivalent transition for im-mobile elements interacting through network connections.For the Kosterlitz-Thouless transition of the XY-model, wepresent a novel analysis based on the statistics of vortextrajectories in space-time, which illustrates the unbindingtransition.

Cristian HuepeDepartment of Engineering Sciences and AppliedMathematicsNorthwestern [email protected]

PP0

Electrokinetic Effect on Electrolytic Flow in Mi-

DS05 Abstracts 191

crochannels

Due to the size effect of the microchannels the electrolyticphenomenon is dominant which is negligible in bulk flow.In the present work an analytic solution to electrokineticflow in a parallel plate microchannel has been developed.The Navier-Stokes equations have been modified to takeinto account the electro-viscous effect. The effect of elec-trokinetics on rate of boundary layer growth is discussed.The equations for fully developed friction factor and Nus-selt number have been developed and interpreted in thephysical domain.

Abhishek JainDepartment of Mechanical, Aerospace and NuclearEngineering,Rensselaer Polytechnic Institute, Troy, New Yorkabhijain [email protected]

PP0

Noise Effect on Dynamic System of a Two Dimen-sional Stoichiometric Discrete Producer-GrazerModel

The grazer can become extinct while having plenty of pro-ducer in a completely deterministic system. An explana-tion for this lies in the bad nutritional quality of the pro-ducer that precludes the grazer from efficiently convertingthe consumed food into its own biomass. In experiments,data are collected on discrete time intervals and it is ob-served that many prey in nature have non-overlapping gen-erations. In our paper, we use a non-overlapping discretemodel to see how producer quality can pull the systems outof oscillations and how it halts chaotic dynamics. We dis-cuss the behavior of our discrete dynamic system: in whichcondition the grazer becomes extinct but the producer stillexhibits chaotic behavior and in which condition the grazerbecomes extinct but the producer has stable equilibrium.Species diversity in nature is accomplished by coexistence.Utilizing a realistic model that consists of two interact-ing species producer and grazer, we discovered a stochasticphenomenon where noise can enhance the coexistence andthereby promote species diversity. We use scaling-law andphase plane to explain this interesting phenomenon.

Yun KangMathematics and StatisticsArizona State [email protected]

PP0

Calcium Waves Mediated Calcium Alternans inCardiac Cells

Contraction of heart cells occurs when calcium ions en-ter the cell through L-type calcium channels, and triggercalcium release from the sarcoplasmic reticulum (SR) bythe opening of ryanodine receptors embedded on the SRmembrane surface. This process is known as calcium in-duced calcium release (CICR). Calcium alternans can bethe result of unstable CICR. Another example of unstableCICR is calcium waves. By using mathematical model, wesuggest that calcium waves can cause calcium alternans.


Young-Seon Lee

Department of MathematicsUniversity of [email protected]

PP0

Alternative Determinism Principle for TopologicalAnalysis of Chaos

The topological analysis of chaos based on a knot-theoreticcharacterization of unstable periodic orbits can only be ap-plied to three-dimensional systems. Still, its core princi-ples, determinism and continuity, apply in any dimension.We propose an alternative framework where these prin-ciples are enforced on triangulated surfaces rather thancurves. As a first step towards a formalism applicable inhigher dimensions, we show that our approach simplifiessignificantly the computation of topological entropies ofthree-dimensional periodic orbits.

Marc LefrancPhLAM/Universite Lille I,[email protected]

Michel NizetteUniversite Libre de BruxellesTheoretical Nonlinear Optics [email protected]

Pierre-Emmanuel MorantPhLAM/Universite Lille [email protected]

PP0

Homogeneous Three-Cell Networks

Coupled cell systems are networks of differential equations.The architecture of a coupled cell network is a graph indi-cating which cells are identical and which cells are coupledto which. We show that there are 34 homogeneous three-cell networks with each cell having at most two inputs asopposed to only two such two-cell networks. We classifycodimension-one synchrony-breaking bifurcations in thesenetworks, showing some surprising features.


Maria LeiteUniversity of [email protected]

PP0

Dynamic and Steady State Analysis of the GastricMucus Gel

Existence of gels in biological systems requires reconsider-ation of standard modeling techniques. The stomach con-tains a gel mucus layer that is responsible for protectingthe stomach from autodigestion. To supply this protec-tion, the gel layer maintains a large gradient in pH. Wediscuss the mechanisms responsible for this phenomena atsteady state. Also, we consider the dynamic response ofthe gel to a surge of acid in the stomach corresponding to

192 DS05 Abstracts

the presence of food.


Frank LynchUniversity of UtahDepartment of [email protected]

PP0

Modeling the Circadian Rhythm of Melatonin Se-cretion and Metabolism

We have measured plasma melatonin concentration and to-tal urinary output of the metabolite for three days in con-stant lighting conditions. The secretion and metabolism ofthe melatonin are modeled with a pharmacokinetic model.The onswitch and offswitch of the secretory pulse are mod-elled as the nonlinear transform of a ”clock”. We comparethree clocks: cosine with 24-hour period; cosine with esti-mated period; van der Pol oscillator outside its limit cycle.

Matthew R. MarlerUniversity of California at San DiegoDepartment of [email protected]

Dan Kripke, Jeffrey ElliottUCSD [email protected], [email protected]

PP0

Recurrence Plot Extension for Spatial Data

Classically introduced recurrence plots (RPs) can only beapplied to one-dimensional data like phase space vectorsand time series. We develop an extended and general-ized RP approach which enables to analyze spatial (higher-dimensional) data regarding recurrent structures. Result-ing RPs have higher dimensions (eg 4). Hence, the mea-sures used to evaluate classic RPs are extended to assesshigher-dimensional recurrence plots. Developed approachis applied to assess bone structure from 2D pQCT imagesof human proximal tibia. (ESA-Project contract #14592)

Norbert MarwanInstitute of Physics,University of [email protected]

PP0

Functions and Formation of Circulation Networkin An Amoeboid Organism

To evaluate performance in a complex survival task,we studied the morphology of transportation networkin an amoeboid organism (true slime mold Physarumpolycehalum) when presented with multiple separate foodsources. The organism comprises a network of tubular el-ements through which intracellular signals and the viscoussol are transported and circulated. I report functions ofthe network shape and mathematical model for networkformation.

Yasumasa NishiuraRIES, Hokkaido University

[email protected]

Toshiyuki NakagakiResearch Institute for Electronic ScienceHokkaido [email protected]


Atsushi TeroHiroshima [email protected]

Tetsuo UedaHokkaido [email protected]

PP0

A Developmental Model of Ocular Dominance Col-umn Formation on a Growing Cortex

We derive an activity-based developmental model of ocu-lar dominance column formation in primary visual cortexthat takes into account cortical growth. The resulting evo-lution equation for the densities of feedforward afferentsfrom the two eyes exhibits a sequence of pattern forminginstabilities as the size of the cortex increases. We use lin-ear stability analysis to investigate the nature of the tran-sitions between successive patterns in the sequence. Weshow that these transitions involve the splitting of existingocular dominance columns, such that the mean width of anOD column is approximately preserved during the course ofdevelopment. This is consistent with recent experimentalobservations of postnatal growth in cat.

Andrew OsterUniversity of [email protected]

PP0

Nonlinear Oscillations in a MicroelectromechanicalSystem

The dynamics of an electret-based, capacitive micro-converter is described by a nonlinear set of ODEs, wherethe equation of a damped, driven oscillator is coupled,through a non linear term, to two first-order, non-lineardifferential equations. The system, which can admit pe-riodic, steady-state solutions, exhibits behaviors typicalof non-linear, Duffing-like oscillators, as jump phenom-ena and hysteretic frequency response curves. In fact, forparticular combinations of the physical parameters of thesystem, multiple steady-state solutions appear. The fre-quency response curves and the stability properties of thesolutions are analyzed with a semianalytic approach. It isalso proved, through perturbative analysis, that the systemalways acts as a linear oscillator under appropriate combi-nations of parameters: in this case the non-linear couplingterm reduces to a viscouslike term, physically interpretableas electromechanical damping.

Fabio Peano, Gianni CoppaPolitecnico di TorinoDipartimento di Energetica

DS05 Abstracts 193


Cristina SerazioPolitecnico di TorinoDipartimento di [email protected]

Antonio D’AngolaUniversita’ degli Studidella [email protected]

Federico PeinettiPolitecnico di TorinoTorino, [email protected]

PP0

Study on Autoresonant Excitation of Non LinearModes in Electron Plasmas

Recent experiments performed at UC Berkeley showed thepossibility of creating large-amplitude synchronized BGKmodes in a pure electron plasma (W. Bertsche et al., Phys-ical Review Letters, 91, 2003). The present work concernsanalytical studies and simulation results aimed at explain-ing the generation mechanisms and the stability of large-amplitude nonlinear modes in non translation-invariantsystems, starting from the theoretical work by L. Fried-land et al. (Physics of Plasmas, 11, 2004).

Fabio Peano, Gianni CoppaPolitecnico di TorinoDipartimento di [email protected], [email protected]

Federico PeinettiPolitecnico di TorinoTorino, [email protected]

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Parameter Study of Nonlinear Subgridscale Modelsfor the Large Eddy Simulation of IncompressibleFlow Problems

In this poster, we examine the effects of various parametersin the numerical simulation of incompressible viscous flowproblems using nonlinear subgridscale models. Specifically,we examine subgridscale models in which artifical viscosityis introduced into the system via a p-Laplacian regulariza-tion, and investigate the dependence of the solution to theparameters p, the order of the p-Laplacian term, and µ,the coefficient of artificial viscosity.

John P. RoopVirginia TechDepartment of [email protected]

PP0

Jet Reaction Control of Nonlinear Wings

We model jet reaction torquer control of chaotic and limit-cycle oscillations (LCO) of aircraft wings. Instead of thediscrete, hinged control surfaces, the vehicle control sys-tems may be implemented with a combination of propul-sive/jet forces. A 2-D nonlinear airfoil model is considered.

The limiter control acts on pitch characteristic of the wing.Its effectiveness to suppress LCO and chaos beyond nom-inal flutter speeds is demonstrated to effectively suppressvibrations and offer gust alleviation.

Erik Bollt, Pier MarzoccaClarkson [email protected], [email protected]

Chrissy RubilloClarkson UniversityMAE [email protected]

PP0

The Dynamic Range of Bursting in a SynapticallyCoupled Network of Square-Wave Bursters

The synchronized bursting of neurons in a certain region ofthe brain stem likely plays a role in the respiratory rhythm.The relevant cells engage in square-wave bursting in theabsence of coupling, if and only if certain experimentallymanipulable parameters are in a subset of their possibleranges. We explain how excitatory synaptic coupling be-tween cells expands this bursting range, by elucidating thedynamical mechanisms underlying transitions from silenceto bursting (symmetric and asymmetric) to tonic spiking(symmetric and asymmetric). This analysis also explainssubtle effects of parameters on burst duration and on otherburst characteristics.

David H. TermanThe Ohio State UniversityDepartment of [email protected]

Jonathan E. RubinUniversity of PittsburghDepartment of [email protected]


Janet Best, Alla BorisyukMathematical Biosciences [email protected], [email protected]

PP0

Mode dynamics in 2D microcavity lasers

We study the lasing dynamics of a stadium-cavity laserby using a model based on the Maxwell and optical-Bloch equations. We numerically investigate the bifurca-tion of stationary lasing states when pumping strength isincreased. A detail analysis is presented for a mode-lockingphenomenon, which yields an asymmetric emission patternin a symmetric cavity.

Satoshi Sunada, Takahisa HarayamaATR Wave Engineering [email protected], [email protected]

Kensuke IkedaRitsumeikan [email protected]

194 DS05 Abstracts

Susumu ShinoharaATR Wave Engineering [email protected]

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Development of Travelling Waves in Weakly Un-stable Shallow Water Systems

Shallow water systems have been observed to develop intoroll waves, which are traveling waves formed after a longtime or distance. To predict these traveling waves, we de-termine the asymptotic behaviour of small disturbances asthey evolve from boundary conditions in a flat inclinedchannel. When the Froude Number is greater than twothe system is unstable when disturbed from uniform flow.For the weakly unstable problem, we determine the evo-lution of the solution over space by applying perturbationtheory. It is found that the solution is dominated by theevolution of the solution along one characteristic. We alsoshow the shock conditions governing the asymptotic solu-tion and use these conditions to determine the approximateshape of the resulting travelling wave, or roll wave, fromthe solution. We present both asymptotic and numericalresults for periodic disturbances in transition, as well asthe travelling wave solutions for the shallow water and theasymptotic evolution equations. Finally, we present an ap-plication of these results to the problem of transition toslug flow in channels.

Richard SpindlerUniversity of VermontDept. of Math and [email protected]

Jun YuUniversity of [email protected]

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Large Linewidth Enhancement Factor in a Mi-crochip Laser

We evidence experimentally that the linewidth enhance-ment factor alpha can take a rather large value (aroundunity) for a non semiconductor laser like a Nd:YAG mi-crochip laser. This raises the hope to observe a rich dy-namic when the laser is subjected to external feedback butin a much direct way than with semiconductor lasers sincethe characteristic frequencies are much lower for microchiplasers (MHz instead of Ghz).

Christophe Szwaj

Universite de Lille (France)Laboratoire [email protected]

Eric LacotLab. SpectroUJF-Grenoble (France)[email protected]

Olivier HugonLab. [email protected]

PP0

Pattern formation in plamodial slime mold onhard/soft surface

Morphology of the plasmodial slime mold, Physarum poly-cephalum, in various media was investigated. Plasmodiumshows disk-like or dendrite shape under rich nutrient orharmful environment, respectively. Interestingly, the samesituation occurred on hard/soft surface of agar mediumand plasmodium prefer hard surface without nutrient tosoft surface with small amount of nutrient. Contact anglesof plasmodium were smaller on harder surface. Plasmod-ium moves in direction of small contact angles irrespectiveof existence of food.

Atsuko TakamatsuDepartment of Electrical Engineering and BioscienceWaseda Universityatsuko [email protected]

PP0

Pulsating Outbreaks in An Sir-Epidemic Modelwith Temporary Immunity

The SIR-epidemic model considers that recovered individ-uals are permanently immune, while the SIS model consid-ers recovered individuals to be immediately re-susceptible.We study the case of temporary immunity in an SIR basedmodel with delayed coupling between the susceptible andremoved classes. We perform a numerical and analyticalbifurcation analysis of the resulting DDE and describe howtemporary immunity leads to recurrent outbreaks and howmodel parameters affect the severity and period of the out-breaks.

Thomas W. CarrSouthern Methodist UniversityDepartment of [email protected]

Michael TaylorDepartment of MathematicsSouthern Methodist [email protected]

PP0

Dendritic cable with active spines: a modellingstudy in the Spike-Diffuse-Spike framework

The spike-diffuse-spike (SDS) model describes a passivedendritic tree with active dendritic spines. Spine head dy-namics is modeled with a simple integrate-and-fire process,whilst communication between spines is mediated by thecable equation. We develop a computational frameworkthat allows the study of multiple spiking events in a net-work of such spines embedded on a simple one-dimensionalcable. In the first instance this system is shown to sup-port saltatory waves with the same qualitative features asthose observed in a model with Hodgkin-Huxley kineticsin the spine head. Moreover, there is excellent agreementwith the analytically calculated speed for a solitary salta-tory pulse. Upon driving the system with time-varyingexternal input we find that the distribution of spines canplay a crucial role in determining spatio-temporal filteringproperties. In particular, the SDS model in response toperiodic pulse train shows a positive correlation betweenspine density and low-pass temporal filtering that is con-sistent with the experimental results of Rose and Fortune(J Neurosci 1997, 17; J Experiment Biol 1999, 202). Fi-

DS05 Abstracts 195

nally we demonstrate one of the ways to incorporate noisethat arises in the spine-head.

Gabriel J. LordHeriot-Watt [email protected]

Stephen CoombesUniversity of [email protected]

Yulia TimofeevaHeriot-Watt [email protected]

PP0

A Collective Motion Algorithm for Tracking Time-Dependent Boundaries

We present a numerical method that allows a formationof communicating agents to follow a concentration gradi-ent and track the boundary of this concentration surface.The algorithm, allows the agents to move in space in anon-stationary environment. Our method is applicable tostudying motions of swarms in biology, as well as to engi-neering applications where boundary detection is an issue.


Ioana A. TriandafNaval Research LaboratoryPlasma Physics [email protected]

PP0

A Karhunen-Loeve Method of CharacterizingSpatio-Temporal Chaos

We present a method of characterizing chaos in complexspatio-temporal patterns, that allows distinguishing be-tween a complex pattern which is involved and could ap-pear chaotic, yet consisting of a variety of coexisting peri-odic or quasiperiodic motions and a spatio-temporal pat-tern trully chaotic. We exemplify our method on a sys-tem of four globally-coupled Ginzburg-Landau equationsderived from the weak electrolyte model for the electro-convection of nematic liquid crystals.


Ioana A. TriandafNaval Research LaboratoryPlasma Physics [email protected]

Iuliana OpreaNaval Research Laboratorytba

PP0

Stabilization of Pulses with Algebraic Decay

The Ginzburg-Landau equation has various unstable soli-tary pulse solutions, which have, however, been observedin systems with two competing instability mechanisms. Insuch systems, the Ginzburg-Landau equation is coupled toa diffusion equation. We use an Evans function method toshow that the effect of the slow diffusion can indeed stabi-lize a pulse when higher-order nonlinearities are taken intoaccount. Current work adapts this approach to the casewhere the diffusion equation has a neutrally stable mode,introducing pulses that decay algebraically rather than ex-ponentially.

Nienke ValkhoffKdV InstituteUniversity of [email protected]

PP0

Structured Pseudospectra in Structural Engineer-ing

The stability of a system is usually studied by analysing thespectrum of an associated linear operator. More accurateinformation, however, can often be obtained by consideringpseudospectra of the operator. We discuss applications ofpseudospectra to stability and vulnerability problems instructural engineering. In particular, we focus on a novelmethod for the computation of pseudospectra with respectto a class of structured perturbations.

Thomas WagenknechtBristol Laboratory for Advanced Dynamics [email protected]

PP0

Floer Homology for Braids on the Two-Disc

We are interested in 1-periodic solutions/periodic points ofHamiltonian systems on the two-disc. Such solutions canbe viewed as braids. For various types of braids we candefine an invariant via Floer homology. The latter can becomputed using the standard Conley index. Applying thistool allows us to obtain various types of forcing results forperiodic solutions.

Jan Bouwe Van Den BergVU AmsterdamDepartment of [email protected]

Wojciech T. WojcikDepartment of MathematicsVrije Universiteit [email protected]

Robert VandervorstVU AmsterdamDepartment of [email protected]

PP0

Detection of Symmetric Homoclinic Orbits toSaddle-Centers in Reversible Systems

We study the persistence of symmetric homoclinic orbits

196 DS05 Abstracts

to saddle-centers in reversible systems under small pertur-bations and develop a perturbation method for detectingsuch orbits. To illustrate the theory we give an examplefor a four-dimensional system arising from a model of anonlinear-optical medium with both quadratic and cubicnonlinearities. Homoclinic orbits in the ODE system cor-respond to embedded solitons, i.e., isolated solitary waveswhich reside in the continuous spectrum of the linearizedsystem and whose tail amplitudes exactly vanish, in thePDE optical model.


PP0

Classical Invariant Theory in Normal Form Com-putations

The normal form for a dynamical system x = F (x) in thevicinity of a fixed point x = 0 is completely characterizedby the polynomial solutions to the Homological equation

AP (x) −Ax · ∇P (x) = 0

where A = ∇F (0) is the linearization at the fixed point.When A is nilpotent, there is an obvious connection be-tween the Homological operator and Classical invarianttheory, which can be exploited to obtain a polynomial rep-resentation of the normal form to all orders. We extendthis approach to include cases where A has a nonzero semi-simple part. Using these ideas along with Grobner basistechniques, we determine the normal forms for all the codi-mension 2 bifurcations with one reversing symmetry.

Shankar C. Venkataramani, Lay May YeapUniversity of ArizonaDepartment of [email protected], [email protected]

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Transient Spatiotemporal Chaos on Complex Net-works

Transient spatiotemporal chaos in a network model basedupon the Gray-Scott cubic autocatalytic reaction-diffusionsystem is examined for different topologies. Motivated byrecent studies on the ”small-world” network topology, ir-regular connections are added to the network’s original reg-ular ring structure. We find that a single added connectioncan significantly decrease the life time of spatiotemporalchaos. A connection spanning a relatively small portionof the entire network (< 15%), however, tends to increasethe system’s transient time, an effect that is explained viathe added connection’s effect on the dynamics local to it.Finally, we find that the addition of two connections cantransform the system’s spatiotemporal chaos from tran-sient to asymptotic.

Renate A. WackerbauerUniversity of Alaska FairbanksDepartment of [email protected]

Safia G. YonkerUniversity of Alaska, FairbanksPhysics [email protected]

ds05 abstracts · 2018. 5. 25. · short-strand dna renaturation by an active micro-mixer...

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