encyclopedia of nonlinear science - princeton...
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e n c y c l o p e d i a o f
nonlinearscience
e n c y c l o p e d i a o f
nonlinearscience
Alwyn Scott, Editor
R O U T L E DG EAn Imprint of Taylor & Francis Group
New York London
Published in 2004 by
RoutledgeAn Imprint of the Taylor & Francis Group29 West 35th StreetNew York, NY 10001www.routledge-ny.com
Published in Great Britain by Routledge11 New Fetter LaneLondon EC4P 4EEwww.routledge.co.uk
Routledge is an imprint of the Taylor & Francis Group
Copyright (c) 2004 by Taylor & Francis Books, Inc.
All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic,mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any informationstorage and retrieval system, without permission in writing from the publisher.
10 9 8 7 6 5 4 3 2 1
Library ofCongressCataloging-in-PublicationDataTo ComeISBN 1-57958-385-7
Contents
Introduction vii
List of Advisers xiii
List of Contributors xv
Alphabetical Entry List xxxiii
Thematic Entry List xxxix
Entries A to Z 000
Index 000
Introduction
Among the several advances of the 20th century,nonlinear science is exceptional for its generality.Although the invention of radio was important forcommunications, the discovery of DNA structurefor biology, the development of quantum theoryfor theoretical chemistry, and the invention ofthe transistor for computer engineering, nonlinearscience is significant in all these areas and manymore. Indeed, it plays a key role in almost everybranch of modern research, as this Encyclopedia ofNonlinear Science shows.In simple terms, nonlinear science recognizes
that the ��whole is more than a sum of its parts,��providing a context for consideration of phenomenalike tsunamis (tidal waves), biological evolution,atmospheric dynamics, and the electrochemicalactivity of a human brain, among many others.For a research scientist, nonlinear science offersnovel phenomena, including the emergence ofcoherent structures (an optical soliton, e.g., ora nerve impulse) and chaos (characterized bythe difficulties in making accurate predictions forsurprisingly simple systems over extended periodsof time). Both these phenomena can be studiedusing mathematical methods described in thisEncyclopedia. From amore fundamental perspective,a wide spectrum of applications arises becausenonlinear science introduces a paradigm shift inour collective attitude about causality. What is thenature of this shift?Consider the difference between linear and non-
linear analyses. Linear analyses are characterized bythe assumption that individual effects can be unam-biguously traced back to particular causes. In otherwords, a compound cause is viewed as the linear (oralgebraic) sum of a collection of simple causes, each
of which can be uniquely linked to a particulareffect. The total effect responding to the total causeis then considered to be just the linear sum of theconstituent effects.A fundamental tenet of nonlinear science is
to reject this convenient, but often unwarranted,assumption. Of course, the notion that componentsof complex causes can interact among themselvesis not surprising to any thoughtful person whomanages to get through an ordinary day ofnormal life, and it is not at all new. Twenty-fivecenturies ago,Aristotle described four types of cause(material, efficient, formal, and final), which overlapand intermingle in ways that were often overlookedin 20th-century thought but are now under scrutiny.Consider some examples of linear scientific thinkingthat are presently being reevaluated in the contextof nonlinear science.--- Around the middle of the 20th century,
behavioral psychologists adopted the theoreticalposition that human mental activity can be reducedto a sum of individual responses to specificstimuli that have been learned at earlier stagesof development. Current research in neuroscienceshows this perspective to be unwarranted.--- Some evolutionary psychologists believe that
particular genes, located in the structure of DNA,can always be related in a one-to-one mannerto individual features of an adult organism,leading to hunts for a ��crime gene�� that seemsabhorrent to moralists. Nonlinear science suggeststhat the relation between genes and features of anadult organism are more intricate than the linearperspective assumes.--- The sad disintegration of space shuttle
Columbia on the morning of February 1, 2003 set off
vii
Introduction
a search for ��the cause of the accident,�� ignoringAristotelian insights into the difficulties of definingsuch a concept, never mind sorting out the pieces.Did themishap occur because the heat-resistant tileswere timeworn (a material cause)? Or because 1.67pounds of debris hit the left wing at 775 ft/s duringtakeoff (an efficient cause)? Perhaps a managementculture that discounted the importance of safetymeasures (a formal cause) should shoulder some ofthe blame.--- Cultural phenomena, in turn, are often viewed
as the mere sum of individual psychologies,ignoring the grim realities of war hysteria andlynch mobs, not to mention the ��tulip craze�� of17th-century Holland, the more recent ��dot-combubble,�� and the outbreak of communal mourningover the death of Princess Diana.
Evolution of the ScienceAs the practice of nonlinear science involves suchabstruse issues, one might expect its history to becheckered, and indeed it is. Mathematical physicsbegan with the 17th-century work of Isaac Newton,whose formulation of the laws ofmechanicalmotionand gravitation explained how the Earth movesabout the Sun, replacing a final cause (God’s plan)with an efficient cause (the force of gravity). Becauseit assumed that the net gravitational force actingon any celestial body is the linear (vector) sumof individual forces, Newton’s theory providessupport for the linear perspective in science,as has often been emphasized. Nonetheless, themathematical system Newton developed (calculus)is the natural language for nonlinear science, and heused this language to solve the two-body problem(collective motion of Earth and Moon)---the firstnonlinear system to be mathematically studied.Also in the 17th century, Christiaan Huygens notedthat two pendulum clocks (which he had recentlyinvented) kept exactly the same time when hangingfrom a common support. (Confined to his room, byan indisposition, Huygens observed the clocks overa period of several days, during which the swingingpendula remained in step.) If the clocks wereseparated to opposite sides of the room, one lostseveral seconds a daywith respect to the other. Fromsmall vibrations transmitted through the commonsupport, he concluded, the two clocks becamesynchronized---a typical nonlinear phenomenon.In the 18th century, Leonhard Euler used
Newton’s laws of motion to derive nonlinear fieldequations for fluid flow, which were augmented acentury later by Louis Navier and George Stokesto include the dissipative effects of viscosity that
are present in real fluids. In their generality,these equations defied solution until the middleof the 20th century when, together with thedigital computer, elaborations of the Navier--Stokes equations provided a basis for generalmodels of the Earth’s atmosphere and oceans,with implications for the vexing question ofglobal warming. During the latter half of the19th century, however, special analytic solutionswere obtained by Joseph Boussinesq and relatedto experimental observations of hydrodynamicsolitary waves by John Scott Russell. These studies---which involved a decade of careful observationsof uniformly propagating ��heaps of water�� oncanals and in wave tanks---were among the earliestresearch programs in the area now recognizedas nonlinear science. At about the same time,Pierre Fran�cois Verhulst formulated and solved anonlinear differential equation---sometimes calledthe logistic equation---to model the populationgrowth of his native Belgium.Toward the end of the 19th century, Henri
Poincare returned to Newton’s original theme,presenting a solution of the three-body problemof celestial motion (e.g., a planet with two moons)in a mathematical competition sponsored by theKing of Sweden. Interestingly, a serious error inthis work was discovered prior to its publication,and he (Poincare, not the Swedish king) eventuallyconcluded that the three-body problem cannot beexactly solved. Now regarded by many as the birthof the ��science of complexity,�� this negative resulthad implications that were not widely appreciateduntil the 1960s,whennumerical studies of simplifiedatmospheric models by Edward Lorenz showedthat nonlinear systems with as few as threedegrees of freedom can readily exhibit the nonlinearphenomenon of chaos. (A key observation here wasof an unanticipated sensitivity to initial conditions,popularly known as the ��butterfly effect�� fromLorentz’s speculation that ��the flap of a butterfly’swings in Brazil [might] set off a tornado in Texas.��)During the first half of the 20th century,
the tempo of research picked up. Although stillcarried on as unrelated activities, there appeareda notable number of experimental and theoreticalstudies now recognized as precursors of modernnonlinear science. Among others, these includeAlbert Einstein’s nonlinear theory of gravitation;nonlinear field theories of elementary particles(like the recently discovered electron) developedby Gustav Mie and Max Born; experimentalobservations of local modes in molecules byphysical chemists (for which a nonlinear theorywas developed by Reinhard Mecke in the 1930s,
viii
Introduction
forgotten, and then redeveloped in the 1970s);biological models of predator--prey populationdynamics formulated by Vito Volterra (to describeyear-to-year variations in fish catches from theAdriatic Sea); observations of a profusion oflocalized nonlinear entities in solid-state physics(including ferromagnetic domain walls, crystaldislocations, polarons, and magnetic flux vorticesin superconductors, among others); a definitiveexperimental and theoretical study of nerve impulsepropagation on the giant axon of the squid byAlan Hodgkin and Andrew Huxley; Alan Turing’stheory of pattern formation in the developmentof biological organisms; and Boris Belousov’sobservations of pattern formation in a chemicalsolution, which were at first ignored (underthe mistaken assumption that they violated thesecond law of thermodynamics) and later confirmedand extended by Anatol Zhabotinsky and ArtWinfree. Just as the invention of the laser in theearly 1960s led to numerous experimental andtheoretical studies in the new field of nonlinearoptics; thus, the steady increases in computingpower throughout the second half of the 20thcentury enabled ever more detailed numericalstudies of hydrodynamic turbulence and chaos,whittling away at the long-established Navier--Stokes equations and confirming the importanceof Poincare’s negative result on the three-bodyproblem.Thus, it was evident by 1970 that nonlinearity
manifests itself in several remarkable properties ofdynamical systems, including the following. (Thereare others, some no doubt waiting to be discovered.)--- Many nonlinear partial differential equations
(wave equations, diffusion equations, and morecomplicated field equations) are often observedto exhibit localized or lump-like solutions, similarto Russell’s hydrodynamic solitary wave. These��coherent structures�� of energy or activity emergefrom initial conditions as distinct dynamic entities,each having its own trajectory in space-time andcharacteristic ways of interacting with others. Thus,they are ��things�� in the normal sense of the word.Interestingly, it is sometimes possible to computethe velocity of emergent entities (their speeds andshapes) from initial conditions and express themas tabulated functions (theta functions or ellipticfunctions), thereby extending the analytic reach ofnonlinear analysis. Examples of emergent entitiesinclude tornados, nerve impulses, magnetic domainwalls, tsunamis, optical solitons, Jupiter’s Great RedSpot, black holes, schools of fish, and cities, to namebut a few. A related phenomenon, exemplified bymeandering rivers, bolts of lightning, andwoodland
paths, is called filamentation, which also causesspotty output beams in poorly designed lasers.--- Surprisingly simple nonlinear systems
(Poincare’s three-body problem is the classic exam-ple) are found to have chaotic solutions, which re-main within a bounded region, while the differencebetween neighboring solution trajectories grows ex-ponentially with time. Thus, the course of a solutiontrajectory is strongly sensitive to its initial conditions(the ��butterfly effect��). Chaotic solutions arise inboth energy-conserving (Hamiltonian) systems andin dissipative systems, and they are fated to wan-der unpredictably as trajectories that cannot be accu-rately extended into the future for unlimited periodsof time. As Lorenz pointed out, the chaotic behaviorof the Earth’s atmosphere makes detailed meteoro-logical predictions problematic, to the delight of themathematician and the despair of the weatherman.Chaotic systems also exhibit ��strange attractors�� inthe solution space, which are characterized by frac-tal (non-integer) dimensions.--- Nonlinear problems often display threshold
phenomena, meaning that there is a relativelysharp boundary across which the qualitative natureof a solution changes abruptly. This is the basicproperty of an electric wall switch, the triggerof a pistol, and of the flip-flop circuit that acomputer engineer uses to store a bit of information.(Indeed, a computer can be viewed as a large,interconnected collection of threshold devices.)Sometimes called ��tipping points�� in the context ofsocial phenomena, thresholds are an important partof our daily experience, where they complicate therelationship of causality to legal responsibility. Wasit the last straw that broke the camel’s back? Or didall of the straws contribute to some degree? Shouldeach be blamed according to its weight? How doesone assign culpability for the ��Murder on the OrientExpress��?--- Nonlinear systems with several spatial coor-
dinates often exhibit spontaneous pattern forma-tion, examples of which include fairy rings of mush-rooms, oscillatory patterns of heart muscle activityunder fibrillation (leading to sudden cardiac arrest),weather fronts, the growth of form in a biologicalembryo, and the Gulf Stream. Such patterns can bechaotic in time and regular in space, regular in timeand chaotic in space, or chaotic in both space andin time, which in turn is a feature of hydrodyamicturbulence.--- If the input to (or stimulation of) a nonlinear
system is a single frequency sinusoid, the output (orresponse) is nonsinusoidal, comprising a spectrumof sinusoidal frequencies. For lossless nonlinearsystems, this can be an efficientmeans for producing
ix
Introduction
energy at integer multiples of the driving frequency,through the process of harmonic generation. Inelectronics, this process is widely used for digitaltuning of radio receivers. Taking advantage of thenonlinear properties of certain transparent crystals,harmonic generation is also employed in laseroptics to create light beams of higher frequency: forexample, conversion of red light to blue.--- Another nonlinear phenomenon is the syn-
chronization of weakly coupled oscillators, first ob-served by the ailing Huygens in the winter of 1665.Now recognized in a variety of contexts, this effectcrops up in the frequency locking of electric powergenerators tied to the same grid and the coupling ofbiological rhythms (circadian rhythms in humans,hybernation of bears, and the synchronized flashingof Indonesian fireflies), in addition to many appli-cations in electronics. Some suggest that neuronalfirings in the neocortex may be mutually synchro-nized.--- Shock waves are familiar to most of us as the
boom of a jet airplane that has broken the soundbarrier or the report of a cannon. Closely relatedfrom a mathematical perspective are the bow waveof a speedboat, the breaking of onshore surf, andthe sudden automobile pileups that can occur on ahighway that is carrying traffic close to itsmaximumcapacity.--- More complicated nonlinear systems can be
hierarchical in nature. This comes about whenthe emergence of coherent states at one levelprovides a basis for new nonlinear dynamics at ahigher level of description. Thus, in the course ofbiological evolution, chemical molecules emergedfrom interactions among the atomic elements, andbiological molecules then emerged from simplermolecules to provide a basis for the dynamicsof a living cell. From collections of cells, multi-cellular organisms emerged, and so on up theevolutionary ladder to creatures like ourselves,who comprise several distinct levels of biologicaldynamics. Similar structures are observed in theorganization of coinage and of military units,not to mention the hierarchical arrangement ofinformation in the human brain.Often, qualitatively related behaviors---involving
one or more of such nonlinear manifestations---arefound in models that arise from different areasof application, suggesting the need for interdisci-plinary communications. By the early 1970s, there-fore, research in nonlinear science was in a statethat the physical chemists might describe as ��su-persaturated.�� Dozens of people across the globewere working on one facet or another of nonlin-ear science, often unaware of related studies in tra-
ditionally unrelated fields. During the mid-1970s,this activity experienced a ��phase change,�� whichcan be viewed as a collective nonlinear effect inthe sociology of science. Unexpectedly, a numberof conferences devoted entirely to nonlinear sci-ence were organized, with participants from a vari-ety of professional backgrounds, nationalities, andresearch interests eagerly contributing. Solid-statephysicists began to talk seriously with biologists,neuroscientists with chemical engineers, and mete-orologists with psychologists. As interdisciplinarybarriers crumbled, these unanticipated interactionsled to the founding of centers for nonlinear sci-ence and the launching of several important researchjournals amid an explosion of research activity. Bythe early 1980s, nonlinear science had gained recog-nition as a key component of modern inquiry, play-ing a central role in a wide spectrum of activities.In the terminology introduced by Thomas Kuhn, anew paradigm had been established.
About this BookThe primary aim of this Encyclopedia is to providea source from which undergraduate and gradu-ate students in the physical and biological sciencescan study how concepts of nonlinear science arepresently understood and applied. In addition, itis anticipated that teachers of science and researchscientists who are unfamiliar with nonlinear con-cepts will use the work to expand their intellec-tual horizons and improve their lectures. Finally, itis hoped that this book will help members of theliterate public---philosophers, social scientists, andphysicians, for example---to appreciate the wealth ofnatural phenomena described by a science that doesnot discount the notion of causality.An early step in writing the Encyclopedia was to
choose the entry subjects---a difficult task that wasaccomplished through the efforts of a distinguishedBoard of Advisers (see page vii--viii), with membersfrom Australia, Germany, Italy, Japan, Russia, theUnited Kingdom, and the United States. After muchsifting and winnowing, an initial list of abouta thousand suggestions was reduced to the 439items given on pages xx--xx. Depending on thesubject matter, the entries are of several types. Someare historical or descriptive, while others presentconcepts and ideas that require notations fromphysics, engineering, or mathematics. Althoughmost of the entries were planned to be about athousand words in length, some---covering subjectsof greater generality or importance---are two or fourtimes as long.Of the many enjoyable aspects in editing
this Encyclopedia, the most rewarding has been
x
Introduction
working with those who wrote it---the contributors.The willing way in which these busy peopleresponded to entry invitations and their enthusiasticpreparation of assignments underscores the degreetowhich nonlinear science has become a communitywith a healthy sense of professional responsibility.In every case, the contributors have tried to presenttheir ideas as simply as possible, with a minimumof technical jargon. For a list of the contributorsand their affiliations, see pages • •, from which it isQ:1evident that they come from 25 different countries,emphasizing the international character of nonlinearscience.A proper presentation of the diverse profes-
sional perspectives that comprise nonlinear sciencerequires careful organization of the Encyclopedia,which we attempt to provide. Although each entryis self-contained, the links among them can be ex-plored in several ways. First, the Thematic Liston pages • •, groups entries within several cate-Q:2gories, providing a useful summary of related en-tries through which the reader can surf. Second, theentries have ``See also´´ notes, both within thetext and at the end of the entry, encouraging thereader to browse outwards from a starting node. Fi-nally, the Index contains a detailed list of topicsthat do not have their own entries but are discussedwithin the context of broader entries. If you can-not find an entry on a topic you expected to find,use the Thematic List or Index to locate the title ofthe entry that contains the item you seek. Addition-ally, all entries have selected bibliographies or sug-gestions for further reading, leading to original re-search and textbooks that augment the overview ap-proach to which an encyclopedia is necessarily lim-ited. Although much of nonlinear science evolvedfrom applied mathematics, many of the entries con-tain no equations or mathematical symbols and canbe absorbed by the general reader. Some entries are
necessarily technical, but efforts have been madeto explain all terms in simple English. Also, manyentries have either line diagrams expanding onexplanations given in the text, or photographsillustrating typical examples.The editing of thisEncyclopedia ofNonlinear Science
culminates a lifetime of study in the area, leavingmeindebted to many. First is the Acquisitions Editor,Gillian Lindsey, who conceived of the project,organized it, and carried it from its beginnings inLondon across the ocean to final publication inNew York. Without her dedication, quite simply,the Encyclopedia would not exist. Equally importantto reaching the finished work were the effortsof the advisers, contributors, and referees, who,respectively, planned, wrote, and vetted the work,and to whom I am deeply grateful. On a broadertime--span are colleagues and students from theUniversity of Wisconsin, Los Alamos NationalLaboratories, the University of Arizona, and theTechnical University ofDenmark,withwhom I haveinteracted over four decades. Although far toomanyto list, these collaborations are fondly remembered,and they provide the basis for much of my editorialjudgment. Finally, I express my gratitude for thegenerous financial support of research in nonlinearscience that has been provided to me since the early1960s by theNational Science Foundation (USA), theNational Institutes of Health (USA), the ConsiglioNazionale delle Ricerche (Italy), the EuropeanMolecular Biology Organization, the Departmentof Energy (USA), the Technical Research Council(Denmark), the Natural Science Research Council(Denmark), the Thomas B. Thriges Foundation, andthe Fetzer Foundation.
Alwyn ScottTucson, Arizona 2003
xi
Editorial Advisory Board
Friedrich H. BusseTheoretical Physics, Universitt Bayreuth,Germany
Antonio DegasperisDipartimento di Fisica, Universit degli Studi di Roma”La Sapienza”
William D. DittoApplied Chaos Lab, Georgia Institute of Technology,USA
Chris EilbeckDepartment of Mathematics, Heriot-Watt University,UK
Sergej FlachMax-Planck-Institut fuer Physik komplexer Systeme,Germany
Herman FlaschkaDepartment of Mathematics, University of Arizona,USA
Hermann HakenCenter for Synergetics, University of Stuttgart,Germany
James P. KeenerDepartment of Mathematics, University of Utah
Yuri KivsharNonlinear Physics Group, Australian National University
Yoshiki KuramotoDepartment of Physics, Kyoto University,Japan
Dave McLaughlinCourant Institute of Mathematical Sciences, New YorkUniversity,USA
Lev A. OstrovskyZel Technologies/National Oceanic &Atmospheric Admin-istration,Environmental Technology Laboratory, Boulder, Colorado,and Institute of Applied Physics,Russia
Edward OttInstitute for Research in Electronics and Applied Physics,University of Maryland,USA
Art WinfreeFormerly Department of Ecology and Evolutionary Biol-ogy,University of Arizona,USA
Ludmila V. YakushevichInstitute of Cell BiophysicsRussia
Lai-Sang YoungCourant Institute of Mathematical Sciences,New York University
xiv
List of Contributors
Ablowitz, Mark J.Professor, Department of Applied MathematicsUniversity of Colorado, Boulder, USAAblowitz--Kaup--Newell--Segur (AKNS) system
Aigner, AndreasResearch Associate, Department of EngineeringMathematics, University of Bristol, UKAtmospheric and ocean sciencesNavier--Stokes equationPartial differential equations, nonlinear
Albano, Ezequiel V.Instituto de Investigaciones Fisicoquιmicas Teoricas yAplicadas (INIFTA) University of La Plata, ArgentinaForest fires
Aratyn, HenrikProfessor, Physics DepartmentUniversity of Illinois at Chicago, USADressing method
Aref, HassanDean of Engineering and Reynolds Metals ProfessorVirginia Polytechnic Institute & State University, USABernoulli’s equationChaotic advectionCluster--cluster coagulationHele--Shaw experimentNewton’s laws of motion
Arrowsmith, DavidProfessor, School of Mathematical Sciences Queen MaryUniversity of London, UKSymbolic dynamicsTopology
Athorne, ChristopherSenior Lecturer, Department of MathematicsUniversity of Glasgow, UKDarboux transformation
Bahr, DavidAssistant Professor, College of Engineering &Architecture, Washington State University, USAGlacial flow
Ball, RowenaDepartment of Theoretical PhysicsAustralian National University, AustraliaFairy rings of mushroomsKolmogorov cascadeSingularity theory
Barnes, HowardUnilever Research Professor of Industrial RheologyDepartment of Mathematics,University of Wales Aberystwyth, WalesRheology
Barthes, MarietteGroupe de Dynamique des Phases Condensees UMRCNRS 5581, Universite Montpellier 2, FranceRayleigh and Raman scattering and IR absorption
Beck, ChristianReader in Applied MathematicsSchool of Mathematical SciencesQueen Mary & Westfield College, UKFree energyMultifractal analysisString theory
xv
List of Contributors
Benedict, KeithSenior Lecturer, School of Physics and AstronomyUniversity of Nottingham, UKAnderson localizationFrustration
Berge, LucProfessor, Commissariat a l’Energie AtomiqueBruyeres FranceDevelopment of singularitiesFilamentationKerr effect
Bernevig, Bogdan A.Physics DepartmentMassachusetts Institute of Technology, USAHolons
Biktashev, VadimLecturer in Applied Maths, Mathematical SciencesUniversity of Liverpool, UKVortex dynamics in excitable media
Binczak, StephaneLaboratoire d’Electronique, Informatique et ImageUniversite de Bourgogne, FranceEphaptic couplingMyelinated nerves
Biondini, GinoAssistant Professor, Department of MathematicsOhio State University, USAEinstein equationsHarmonic generation
Blair, DavidProfessor, School of PhysicsThe University of Western Australia, AustraliaGravitational waves
Boardman, Alan D.Professor of Applied PhysicsInstitute for Materials ResearchUniversity of Salford, UKPolaritons
Bollt, ErikProfessor, Department of MathUnited States Naval Academy, USAMarkov partitionsOrder from chaos
Boon, J.-P.Professor, Faculte des SciencesUniversite Libre de Bruxelles, BelgiumLattice gas methods
Borckmans, PierreCenter for Nonlinear Phenomena & Complex SystemsUniversite Libre de Bruxelles, BelgiumTuring patterns
Boumenir, AminDepartment of MathematicsState University of West Georgia, USAGel’fand--Levitan theory
Bountis, TassosProfessor, Department of Mathematics andCenter for Research and Application of NonlinearSystemsUniversity of Patras, GreecePainleve analysis
Boyd, Robert W.Professor, The Institute of OpticsUniversity of Rochester, USAFrequency doubling
Bradley, ElizabethAssociate Professor, Department of Computer ScienceUniversity of Colorado, USAKirchhoff’s laws
De Bruyn, JohnProfessor, Department of Physics andPhysical Oceanography, Memorial University ofNewfoundland, CanadaPhase transitionsThermal convection
Bullough, RobinProfessor, Applied MathematicsUniversity of Manchester Institute of Science andTechnology, UKMaxwell--Bloch equationsSine-Gordon (SG) equation
Bunimovich, LeonidRegents Professor, Department of MathematicsGeorgia Institute of Technology, USABilliardsDeterministic walks in random environmentsLorentz gas
xvi
List of Contributors
Busse, Friedrich (Adviser)Professor, Theoretical PhysicsUniversitat Bayreuth, GermanyDynamos, homogeneousFluid dynamicsMagnetohydrodynamics
Calini, Annalisa M.Associate Professor, Department of MathematicsCollege of Charleston, USAElliptic functionsMelnikov methodJump phenomena
Caputo, Jean GuyLaboratoire de Mathematiques, Institut National desSciences Appliquees de Rouen, FranceJump phenomena
Censor, DanProfessor, Department of Electrical and ComputerEngineering, Ben-Gurion University of the Negev, IsraelVolterra series and operators
Chen, Wei-YinProfessor, Department of Chemical EngineeringUniversity of Mississippi, USAStochastic processes
Chernitskii, Alexander A.Department of Physical ElectronicsSt. Petersburg Electrotechnical University, RussiaBorn--Infeld equations
Chiffaudel, ArnaudCEA-Saclay (Commissariat a l’Energie Atomique) &CNRS (Centre National de la Recherche Scientifique)FranceHydrothermal waves
Choudhury, S. RoyProfessor, Department of MathematicsUniversity of Central Florida, USAKelvin--Helmholtz instabilityLorenz equations
Christiansen, PeterProfessor, Informatics and Mathematical ModellingTechnical University of Denmark, DenmarkSeparation of variables
Christodoulides, DemetriosProfessor, CREOL/School of OpticsUniversity of Central Florida, USAIncoherent Solitons
Coskun, TamerResearch Associate, Medical SciencesIndiana University-Purdue University IndianapolisUSAIncoherent solitons
Cruzeiro-Hansson, LeonorHonorary Fellow, Department of MathematicsHeriot-Watt University, UKDavydov soliton
Cushing, JimProfessor, Department of MathematicsUniversity of Arizona, USAPopulation dynamics
Davies, BrianDepartment of MathematicsAustralian National University, AustraliaIntegral transformsPeriod doubling
Davis, William C.Formerly, Los Alamos National LaboratoryUSAExplosions
Deconinck, BernardAssistant Professor, Department of Applied MathsUniversity of Washington, USAKadomtsev--Petviashvili equationPeriodic spectral theoryPoisson brackets
Degallaix, JeromeSchool of PhysicsThe University of Western AustraliaAustraliaGravitational Waves
Deift, PercyProfessor, Department of MathematicsCourant Institute of Mathematical SciencesNew York University, USARandom matrix theory IV: Analytic methodsRiemann--Hilbert problem
Deryabin, Mikhail V.Department of MathematicsTechnical University of Denmark, DenmarkKolmogorov--Arnold--Moser theorem
xvii
List of Contributors
Diacu, FlorinProfessor, Department of Mathematics and StatisticsUniversity of Victoria, CanadaCelestial mechanicsN -body problem
Ding, MingzhouProfessor, Center for Complex Systems andBrain Sciences Florida Atlantic University, USAIntermittency
Degasperis, Antonio (Adviser)Professor, Dipartimento di Fisica, Universita degli Studidi Roma ”La Sapienza,” Italy
Ditto, William D. (Adviser)Applied Chaos Lab, Georgia Institute of Technology, USA
Dolgaleva, KsenaiDepartment of PhysicsM.V. Lomonosov Moscow State UniversityMoscow, Russian FederationFrequency doubling
Donoso, Jose M.Facultad de MatematicasUniversidad ComplutenseMadrid, SpainBall lightning
Doucet, ArnaudSignal Processing Group, Department of EngineeringCambidge University, UKMonte Carlo methods
Dritschel, DavidProfessor, Department of Applied MathematicsThe University of St. Andrews, UKContour dynamics
Dupuis, GerardChimie generale et organiqueLycee Faidherbe de Lille, FranceBelousov--Zhabotinsky reaction
Easton, RobertProfessor, Department of Applied MathUniversity of Colorado at Boulder, USAConley index
Eckhardt, BrunoProfessor, Fachbereich PhysikPhilipps Universitat Marburg, Germany
Lagrangian chaosMaps in the complex planePeriodic orbit theoryQuantum chaosRandom matrix theory I: origins andphysical applications
Shear flowSolar systemUniversality
Eilbeck, Chris (Adviser)Professor, Department of MathematicsHeriot-Watt University, UKDiscrete self-trapping system
Elgin, JohnProfessor, Maths DepartmentImperial College of ScienceTechnology and Medicine, London, UKKuramoto--Sivashinsky equation
Emmeche, ClausAssociate Professor andHead of Center for the Philosophyof Nature and Science StudiesUniversity of Copenhagen, DenmarkCausality
Enolskii, VictorProfessor, Heriot-Watt University, UKTheta functions
Falqui, GregorioLecturer, Mathematical Physics SectorInternational School for Advanced StudiesTrieste, ItalyHodagraph Transform
Falkovich, GregoryProfessor, Department of Physics of Complex SystemsWeizmann Institute of Science, IsraelMixingTurbulence
Faris, WilliamProfessor, Department of MathematicsUniversity of Arizona, USAMartingales
Feddersen, HenrikResearch Scientist, Climate Research DivisionDanish Meteorological Institute, DenmarkForecasting
xviii
List of Contributors
Fedorenko, VladimirSenior Scientific Researcher, Institute of MathematicsNational Academic of Science of Ukraine, UkraineOne-dimensional maps
Fenimore, Paul W.Theoretical Biology and Biophysics GroupLos Alamos National Laboratory, USAProtein dynamics
Flach, Sergej (Adviser)Max-Planck-Institut fur Physik komplexer SystemeGermany.Discrete breathersSymmetry: equations vs. solutions
Flaschka, Hermann (Adviser)Professor, Department of MathematicsUniversity of Arizona, USAToda lattice
Fletcher, NevilleProfessor, Department of ElectronicMaterials EngineeringAustralian National University, AustraliaOvertones
Florιa, Luis MarioDepartment of Theory and Simulation ofComplex SystemsInstituto de Ciencia de Materiales de Aragon, SpainAubry--Mather theoryCommensurate--incommensurate transitionFrenkel--Kontorova model
Forrester, PeterDepartment of Mathematics and StatisticsUniversity of Melbourne, AustraliaRandom matrix theory II: Algebraic developments
Fowler, Beall W.Emeritus Professor, Physics DepartmentLehigh University, USAColor centers
Fraedrich, KlausProfessor, Universitat Hamburg Meteorological InstituteGermanyGeneral circulation models of the atmosphere
Freites, Juan AlfredoDepartment of Physics and AstronomyUniversity of California, Irvine, USAMolecular dynamics
Frieden, RoyOptical Sciences CenterUniversity of Arizona in Tucson, USAInformation theory
Friedrich, J.Professor, Lehrstuhl fur Physik WeihenstephanTechnische Universitat Munchen, GermanyHole burning
Fuchikami, NobukoDepartment of PhysicsTokyo Metropolitan University, JapanDripping faucet
Gallagher, MarcusSchool of Information Technology &Electrical EngineeringThe University of Queensland, AustraliaMcCulloch--Pitts networkPerceptron
Garnier, NicolasLaboratoire de PhysiqueEcole Normale Superieure de Lyon, FranceHydrothermal waves
Gaspard, PierreDepartement de MathematiqueUniversite Libre de Bruxelles, BelgiumEntropyMapsQuantum theoryRossler systems
Glass, LeonProfessor, Department of PhysiologyMcGill University, CanadaCardiac arrhythmias and electrocardiogram
Glendinning, PaulProfessor, Department of MathematicsUniversity of Manchester Institute ofScience and Technology, UKHenon mapInvariant manifolds and setsRoutes to chaos
Goriely, AlainProfessor, Department of MathematicsUniversity of Arizona, USANormal forms theory
xix
List of Contributors
Grand, SteveDirector, Cyberlife Research Ltd., UKArtificial life
Gratrix, SamMaths Department, Imperial College of ScienceTechnology and Medicine, UKKuramoto--Sivashinsky equation
Grava, TamaraMaths DepartmentImperial College of ScienceTechnology and Medicine, UKHodograph transformN -soliton formulasZero-dispersion limits
Grimshaw, RogerProfessor, Department of Mathematical SciencesLoughborough University, UKGroup velocityKorteweg--de Vries equationWater waves
Giuliani, AlessandroIstituto Superiore di Sanita, Rome, ItalyAlgorithmic complexity
Haken, Herman (Adviser)Professor Emeritus, Fakultet fur PhysikUniversity of Stuttgart, GermanyGestalt phenomenaSynergetics
Hallinan, JenniferInstitute for Molecular BioscienceThe University of Queensland, AustraliaGame of lifeGame theory
Hamilton, MarkProfessor, Department of Mechanical EngineeringUniversity of Texas at Austin, USANonlinear acoustics
Hamm, PeterProfessor, Physikalisch-Chemisches InstitutUniversitat Zurich, SwitzerlandFranck--Condon factorHydrogen bondPump-probe measurements
Hasselblatt, BorisProfessor, Department of MathematicsTufts University, USAAnosov and axiom A systemsMeasures Phase space
Hastings, AlanProfessor, Department of Environmental Scienceand PolicyUniversity of California, USAEpidemiology
Hawkins, JaneProfessor, Department of MathematicsUniversity of North Carolina at Chapel Hill, USAErgodic theory
Helbing, DirkInstitute for Economics and TrafficDresden University of Technology, GermanyTraffic flow
Henry, BruceDepartment of Applied MathematicsUniversity of New South Wales, AustraliaEquipartion of energyHenon--Heiles system
Henry, BryanDepartment of Chemistry and BiochemistryUniversity of Guelph, CanadaLocal modes in molecules
Hensler, GerhardProfessor, Institut fur AstronomieUniversitats-Sternwarte Wien, AustriaGalaxies
Herrmann, HansInstitute for Computational PhysicsUniversity of Stuttgart, GermanyDune formation
Hertz, JohnProfessor, Nordic Institute for Theoretical PhysicsDenmarkAttractor neural network (ANN)
Hietarinta, JarmoProfessor, Department of PhysicsUniversity of Turku, FinlandHirota’s method
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List of Contributors
Hill, LarryDetonation Science & TechnologyLos Alamos National Laboratory, USAEvaporation wave
Hjorth, Poul G.Lektor, Department of MathematicsTechnical University of Denmark, DenmarkKolmogorov--Arnold-Moser theorem
Holden, ArunProfessor of Computational BiologySchool of Biomedical SciencesUniversity of Leeds, UKExcitabilityHodgkin--Huxley equationsIntegrate and fire neuronMarkin--Chizmadzhev modelPeriodic burstingSpiral wavesThreshold phenomena
Hommes, CarsProfessor, Center for Nonlinear Dynamics in Economicsand Finance, Department of Quantitative EconomicsUniversity of Amsterdam, The NetherlandsEconomic dynamics
Hone, AndrewLecturer in Applied MathematicsInstitute of Mathematics and StatisticsUniversity of Kent at Canterbury, UKExtremum principlesOrdinary differential equations, nonlinearRiccati equations
Hood, AlanProfessor, School of Mathematical andComputational SciencesUniversity of St Andrews, UKCharacteristics.
Houghton, ConorDublin School of MathematicsTrinity College, IrelandInstantonsYang--Mills theory
Howard, James E.Research Associate, Department of PhysicsUniversity of Colorado at Boulder, USANontwist mapsRegular and chaotic dynamics in atomic physics
Ivey, ThomasDepartment of MathematicsCollege of Charleston, USADifferential geometryFramed space curves
Jimenez, SalvadorProfessor, Departamento de MatematicasUniversidad Alfonso X El SabioMadrid, SpainCharge density waves
Joannopoulos, John D.Professor, Department of PhysicsMassachusetts Institute of Technology, USAPhotonic crystals
Johansson, MagnusDepartment of Physics and Measurement TechnologyLinkoping University, SwedenDiscrete nonlinear Schrodinger equations
Johnson, Steven G.Postdoctoral Associate, Department of PhysicsMassachussetts Institute of Technology, USAPhotonic crystals
Joshi, NaliniProfessor, School of Mathematics and StatisticsUniversity of Sydney, AustraliaSolitons
Kaneko, KunihikoDepartment of Pure and Applied SciencesTokyo University, JapanCoupled map lattice
Kantz, HolgerProfessor, Max-Planck-Institut fu komplexer SystemeGermanyTime series analysis
Keener, James P. (Adviser)Professor, Department of MathematicsUniversity of Utah, USA
Kennedy, Michael PeterProfessor of Microelectronic EngineeringUniversity College, Cork, IrelandChua’s circuit
xxi
List of Contributors
Kevrekidis, Panayotis G.Assistant Professor, Department of Mathematicsand StatisticsUniversity of Massachusetts, Amherst, USABinding energyCollisionsWave of translation
Khanin, KonstantinProfessor, Department of MathematicsHeriot-Watt University, UKDenjoy theory
King, AaronAssistant Professor, Department of Ecologyand Evolutionary BiologyUniversity of Tennessee, Knoxville, USAPhase plane
Kirby, MichaelProfessor, Department of MathematicsColorado State University, USANonlinear signal processing
Kivshar, Yuri (Adviser)Nonlinear Physics GroupAustralian National University, AustraliaOptical fiber communications
Kiyono, KenResearch Fellow of the Japan Society for thePromotion of ScienceEducational Physiology LaboratoryUniversity of Tokyo, JapanDripping Faucet
Knott, RonDepartment of MathematicsUniversity of Surrey, UKFibonacci series
Kocarev, LjupcoAssociate Research ScientistInstitute for Nonlinear ScienceUniversity of CaliforniaSan Diego, USA
Konopelchenko, Boris G.Professor, Dipartimento di FisicaUniversity of Lecce, ItalyMultidimensional solitons
Konotop, Vladimir V.Centro de Fιsica Teorica e ComputacionalComplexo Interdisciplinar da Universidade de LisboaPortugalWave propagation in disordered media
Kosevich, ArnoldPhysics-Engineering Institute of Low TemperaturesKharkov State Polytechnical University, UkraineBreathersDislocations in crystalsEffective massLandau-Lifshitz (LL) equationSuperfluiditySuperlattices
Kovalev, AlexanderInstitute for Low Temperature Physics and EngineeringNational Academy of Sciences of Ukraine, UkraineContinuum approximationsTopological defects
Kramer, PeterAssistant Professor, Department of MathematicalSciences, Rensselaer Polytechnic Institute, USABrownian motionFokker-Planck equation
Krinsky, ValentinProfessor, Institut Non-Lineaire de Nice, FranceCardiac muscle models
Kuramoto, Yoshiki (Adviser)Department of Physics, Kyoto University, JapanPhase dynamics
Kurin, V.Institute for Physics of MicrostructuresRussian Academy of Science, RussiaCherenkov radiation
Kuvshinov, Viatcheslav I.Professor, Institute of PhysicsBelarus Academy of Sciences, BelarusBlack holesCosmological modelsFractalsGeneral relativity
Kuznetsov, VadimAdvanced Research FellowDepartment of Applied MathematicsUniversity of Leeds, UKRotating rigid bodies
xxii
List of Contributors
LaBute, Montiago X.Theoretical Biology and Biophysics GroupLos Alamos National Laboratory, USAProtein structure
Lakshmanan, MuthusamyProfessor, Department of PhysicsBharathidasan University, Tiruchirapalli, IndiaEquations, nonlinearNonlinear electronics
Landa, P.S.Professor, Department of PhysicsMoscow State University, RussiaFeedbackPendulumQuasilinear analysisRelaxation oscillators
Landsberg, PeterProfessor, Faculty of Mathematical StudiesUniversity of Southampton, UKDetailed balance
Lansner, AndersDepartment of Numerical Analysis andComputer Science (NADA)Royal Institute of Technology (KTH), SwedenCell assembliesNeural network models
Lee, JohnDepartment of Mechanical EngineeringMcGill University, Montreal, Quebec, CanadaFlame front
Lega, JocelineProfessor, Department of MathematicsUniversity of Arizona, USAEquilibriumFredholm theorem
Lepeshkin, NickThe Institute of OpticsUniversity of Rochester, USAFrequency doubling
Levi, DecioProfessor, Dipartimento di FisicaUniversita degli Studi di Roma III, ItalyDelay-differential equations
Lewis, KarenHydrologist, SS Papadopulos & Associates, BoulderColorado, USAGlacial flow
Lichtenberg, Allan J.Professor, Department of Electrical Engineering andComputer ScienceUniversity of California at Berkeley, USAArnold diffusionAveraging methodsElectron beam microwave devicesFermi acceleration and Fermi mapFermi--Pasta--Ulam (FPU) oscillator chainParticle acceleratorsPhase space diffusion and correlations
Liley, DavidSchool of Biophysical Sciences andElectrical EngineeringSwinburne University of Technology, AustraliaElectroencephalogram at mesoscopic scales
Lonngren, Karl E.Professor, Department of Electrical andComputer EngineeringUniversity of Iowa, USAPlasma soliton experiments
Losert, WolfgangAssistant Professor, Institute for Plasma ResearchUniversity of Maryland, USAGranular materialsPattern formation
Luecke, ManfredInstitut fur Theoretische PhysikUniversitat des Saarlandes, Saarbrucken, GermanyThermo-diffusion effects
Ma, Wen-XiuProfessor, Department of MathematicsUniversity of South Florida, USAIntegrability
Macaskill, CharlesSchool of Mathematics and StatisticsUniversity of Sydney, AustraliaJupiter’s great red spot
Maggio, Gian MarioCenter for Wireless Communications (CWC)University of California, San Diego, USADamped driven anharmonic oscillator
xxiii
List of Contributors
Maini, Philip K.Professor, Centre for Mathematical BiologyMathematical Institute, University of Oxford, UKMorphogenesis, biological
Mainzer, KlausDepartment of Philosophy of ScienceUniversity of Augsburg, GermanyArtificial intelligenceCellular nonlinear networksDynamical systems
Malomed, Boris A.Professor, Department of Interdisciplinary StudiesFaculty of Engineering, Tel Aviv University, IsraelComplex Ginzburg--Landau (CGL) equationConstants of motion and conservation lawsMultisoliton perturbation theoryNonlinear Schrodinger equationsPower balance
Manevitch, LeonidProfessor, Institute of Chemical Physics, RussiaHeat conductionMechanics of solidsPeierls barrier
Manneville, PaulLaboratoire d’Hydrodynamique (LadHyX)Ecole Polytechnique in Palaiseau, FranceSpatiotemporal chaos
Marklof, JensSchool of Mathematics, University of Bristol, UKCat map
Mart�nez, Pedro JesusDepartment of Theory and Simulation ofComplex SystemsInstituto de Ciencia de Materiales de Aragon, SpainFrenkel--Kontorova model
Masmoudi, NaderAssociate Professor, Department of MathematicsCourant Institute of Mathematical SciencesNew York University, USABoundary layers
Mason, LionelMathematical Institute, Oxford University, UKTwistor theory
Mayer, AndreasInstitute for Theoretical PhysicsUniversity of Regensburg, GermanySurface waves
McKenna, JoeProfessor, Department of MathematicsUniversity of Connecticut, USATacoma narrows bridge collapse
McLaughlin, Kenneth (Adviser)Associate ProfessorDepartment of MathematicsUniversity of North Carolina at Chapel Hill, USARandom matrix theory III: combinatorics
McLaughlin, RichardAssociate Professor, Department of MathematicsUniversity of NorthCarolina, Chapel Hill, USAPlume dynamics
McMahon, BenTheoretical Biology and Biophysics GroupLos Alamos National Laboratory, USAProtein dynamicsProtein Structure
Meiss, JamesProfessor, Department of Applied MathematicsUniversity of Colorado at Boulder, USAHamiltonian systemsStandard mapSymplectic maps
Miura, RobertProfessor, Department of Mathematical SciencesNew Jersey Institute of TechnologyNewark, USANonlinear toys
Moloney, Jerome V.Professor, Department of MathematicsUniversity of Arizona, USANonlinear optics
MLrk, JesterOptoelectronics, Research Center COMTechnical University of DenmarkDenmarkSemiconductor laser
xxiv
List of Contributors
Mornev, OlegProfessor, Institute of Theoretical andExperimental Biophysics, RussiaGeometrical optics, nonlinearGradient systemNerve impulsesZeldovich--Frank-Kamenetsky equation
Mosekilde, ErikProfessor, Department of PhysicsTechnical University of Denmark, DenmarkNephron dynamics
Mueller, Stefan C.Department of BiophysicsOtto-von-Guericke-Universitat Magdeburg, GermanyScroll waves
Mullin, TomProfessor of Physics and Director of Manchester Centrefor Nonlinear DynamicsUniversity of Manchester, UKBifurcationsCatastrophe theoryTaylor--Couette flow
Mygind, JesperProfessor, Department of PhysicsTechnical University of Denmark, DenmarkJosephson junctionsSuperconducting quantum interference device(SQUID)
Nakamura, YoshiharuAssociate Professor, Institute of Space andAstronautical Science, Faculty of Science and TechnologyKeio University, Yokohama, JapanPlasma soliton experiments
Natiello, MarioCentre for Mathematical SciencesLund University, SwedenWinding numbers
Newell, AlanProfessor, Department of MathematicsUniversity of Arizona, USAInverse scattering method or transform
Newton, Paul K.Professor, Department of Aerospace and MechanicalEngineeringUniversity of Southern California, USA
Berry’s phaseChaos vs. turbulence
Neyts, KristiaanProfessor, Department of Electronics andInformation SystemsGhent University, BelgiumLiquid crystals
Nicolis, G.Professor, Faculte des SciencesUniversite Libre de Bruxelles, BelgiumBrusselatorChemical kineticsNonequilibrium statistical mechanicsRecurrence
Nunez, PaulProfessor, Brain Physics GroupDepartment of Biomedical EngineeringTulane University, USAElectroencephalogram at large scales
Olsder, Geert-JanFaculty of Technical Mathematics and InformaticsDelft University of Technology, DenmarkIdempotent analysis
Olver, Peter J.Professor, School of MathematicsUniversity of Minnesota, USALie algebras and Lie groups
Ostrovsky, Lev (Adviser)Environmental Technology LaboratoryZel Technologies/National Oceanic & AtmosphericAdministration, Boulder, Colorado, USA andInstitute of Applied Physics, RussiaHurricanes and tornadoesModulated wavesNonlinear acousticsShock waves
Ott, Edward (Adviser)Distinguished University ProfessorInstitute for Research in Electronics and Applied PhysicsUniversity of Maryland, USA
Palmer, JohnProfessor, Department of MathematicsUniversity of Arizona, USAMonodromy preserving deformations
xxv
List of Contributors
Pascual, Pedro J.Associate ProfessorDepartamento de Ingenieria InformaticaUniversidad Autonoma de Madrid, SpainCharge density waves
Pedersen, Niels FalsigDepartment of Power EngineeringTechnical University of Denmark, DenmarkLong Josephson junctionsSuperconductivity
Pelinovsky, DmitryAssistant Professor, Department of MathsMcMaster University, CanadaCoupled systems of partial differential equationsEnergy analysisGeneralized functionsLinearizationManley--Rowe relationsN -wave interactionsNumerical methodsSpectral analysis
Pelletier, JonAssistant Professor, Department of GeosciencesUniversity of Arizona, USAGeomorphology and tectonics
Pelloni, BeatriceMathematics DepartmentUniversity of Reading, UKBoundary value problemsBurgers’ equation
Petty, MichaelProfessor, Centre for Molecular andNanoscale ElectronicsUniversity of Durham, UKLangmuir--Blodgett films
Peyrard, MichelProfessor of Physics, Laboratoire de PhysiqueEcole Normale Superieure de Lyon, FranceBiomolecular solitons
Pikovsky, ArkadyDepartment of Physics Universitat Potsdam, GermanySynchronizationVan der Pol equation
Pitchford, JonLecturer, Department of Biology, University of York, UKRandom walks
Pojman, John A.Professor and Coordinator of Undergraduate ProgramDepartment of Chemistry and BiochemistryThe University of Southern MississippiHattiesburg, USAPolymerization
Rabinovich, MikhailResident Physicist, Institute for Nonlinear ScienceUniversity of California at San Diego, USAChaotic dynamics
Ranada, Antonio F.Facultad de Fisica, Universidad ComplutenseMadrid, SpainBall lightning
Recami, ErasmoProfessore AssociateFacolta Universita Statale di Bergamo, ItalyTachyons and superluminal motion
Reucroft, StephenProfessor of Physics, Northeastern UniversityBoston, Massachusetts, USAHiggs boson
Ricca, Renzo L.Professor, Dipartimento di Matematica e ApplicazioniUniversita di Milano-Bicocca, Milan, ItalyKnot theoryStructural complexity
Robinson, JamesMathematics InstituteUniversity of Warwick, UKAttractorsDimensionsFunction SpacesFunctional analysis
Robnik, MarkoProfessor, Center for Applied Mathematics andTheoretical PhysicsUniversity of Maribor, SloveniaAdiabatic invariantsDeterminism
Rogers, ColinProfessor, School of MathematicsUniversity of New South Wales, AustraliaBacklund transformations
xxvi
List of Contributors
Romanenko, ElenaSenior Scientific Researcher, Institute of MathematicsNational Academy of Science of Ukraine, UkraineTurbulence, ideal
Rouvas-Nicolis, C.Climatologie DynamiqueInstitut Royal Meteorologique de BelgiqueBelgiumRecurrence
Ruijsenaars, SimonCenter for Mathematics and Computer ScienceThe NetherlandsDerrick--Hobart theoremParticles and antiparticles
Sabatier, PierreProfessor, Physique MathematiqueUniversite Montpellier II, FranceInverse problems
Sakaguchi, HidetsuguDepartment of Applied Science for Electronicsand MaterialsKyushu University, JapanCoupled oscillators
Salerno, MarioDepartment of Physics ”E. R. Caianiello”Universita di Salerno, ItalyBethe ansatzSalerno equation
Sandstede, BjornAssociate Professor, Department of MathematicsOhio State University, USAEvans function
Satnoianu, RazvanDepartment of Mathematics, City University, UKFiffusionReaction diffusion systems
Sauer, TimProfessor, Department of MathematicsGeorge Mason University, USAEmbedding methods
Savin, AlexanderProfessor, Moscow Institute of Physics and TechnologyRussiaPeierls barrier
Schattschneider, DorisProfessor, Department of MathematicsMoravian College, BethlehemPennsylvania, USATessellation
Schmelcher, PeterInstitute for Physical ChemistryUniversity of Heidelberg, GermanyHartree approximation
Scholl, EckehardProfessor, Institut fur Theoretische PhysikTechnische Universitat Berlin, GermanyAvalanche breakdownDiodesDrude modelSemiconductor oscillators
Schuster, PeterInstitut fur Theoretische Chemie undMolekulare Strukturbiologie, AustriaBiological evolutionCatalytic hypercycleFitness landscape
Scott, Alwyn (Editor)Emeritus Professor of MathematicsUniversity of Arizona, USACandleDiscrete self-trapping systemDistributed oscillatorsEmergenceEuler--Lagrange equationsHierarchies of nonlinear systemsLaboratory models of nonlinear wavesLifetimeMatter, nonlinear theory ofMultiplex neuronNeuristorQuantum nonlinearityRotating wave approximation (RWA)Solitons, a brief historyState diagramsSymmetry groupsTachyons and superluminal motionWave packets, linear and nonlinear
Segev, MordechaiProfessor, Technion-Israel Institute of TechnologyHaifa, IsraelIncoherent solitons
xxvii
List of Contributors
Shalfeev, VladimirHead of Department of Oscillation TheoryNizhni Novgorod State University, RussiaParametric amplification
Sharkovsky, AlexanderInstitute of MathematicsNational Academy of Sciences of Ukraine, UkraineOne-dimensional mapsTurbulence, ideal
Sharman, RobertUniversity Corporation for Atmospheric ResearchBoulder, Colorado, USAClear air turbulence
Shinbrot, TroyAssociate ProfessorDepartment of Chemical and Biochemical EngineeringRutgers University, Piscataway, USAControlling chaos
Shohet, LeonProfessor, Department of Electrical andComputer EngineeringUniversity of Wisconsin Madison, USANonlinear plasma waves
Siwak, PawelDepartment of Electrical EngineeringPoznan University of Technology, PolandIntegrable cellular automata
Smil, VaclavProfessor, Department of GeographyUniversity of Manitoba, CanadaGlobal warming
Sobell, HenryIndependent scholar, New York, USADNA premelting
Solari, Hernan GustavoDepartamento FιsicaUniversity of Buenos Aires, ArgentinaLasers
Soljacic, MarinPrincipal Research Scientist, Department of PhysicsMassachusetts Institute of Technology, USAPhotonic crystals
SLrensen, Mads PeterAssociate ProfessorInformatics and Mathematical ModelingTechnical University of Denmark, DenmarkCollective coordinatesMultiple scale analysisPerturbation theory
Sornette, DidierProfessor, Laboratoire de Physique de laMatiere CondenseeUniversite de Nice - Sophia Antipolis, FranceSandpile model
Spatschek, KarlProfessor, Institute for Theoretical Physics 1Heinrich-Heine-Universitat Dusseldorf, GermanyCenter manifold reductionDispersion management
Stauffer, DietrichInstitute for Theoretical PhysicsUniversity of Cologne, GermanyPercolation theory
Stefanovska, AnetaFaculty of Electrical EngineeringUniversity of Ljubljana, SloveniaFlip-flop circuitInhibitionNonlinearity, definition ofQuasiperiodicityWavelets
Storb, UlrichDrittmittelbeschaftigteInstitut fur Experimentelle PhysikOtto-von-Guericke-UniversitatMagdeburg, GermanyScroll waves
Strelcyn, Jean-MarieProfesseur, Departement de MathematiquesUniversite de RouenMont Saint Aignan Cedex, FrancePoincare theorems
Suris, Yuri B.Researcher, Department of MathematicsTechnische Universitat Berlin, GermanyIntegrable lattices
xxviii
List of Contributors
Sutcliffe, PaulReader in Mathematical PhysicsInstitute of Mathematics and StatisticsUniversity of Kent at Canterbury, UKSkyrmions
Swain, JohnProfessor, Department of PhysicsNortheastern University, Boston, USADoppler shiftQuantum field theoryTensors
Tabor, MichaelProfessor, Department of MathematicsUniversity of Arizona, USAGrowth patterns
Tajiri, MasayoshiProfessor, Department of Mathematical SciencesOsaka Prefecture University, JapanSolitons, types ofWave stability and instability
Tass, PeterInstitut fur MedizinForschungszentrum Julich, GermanyStochastic analyses of neural systems
Taylor, RichardAssociate Professor, Materials Science InstituteUniversity of Oregon, USALevy flights
Teman, R.Laboratoire d’Analyse NumeriqueUniversite de Paris Sud, FranceInertial manifolds
Thompson, J.M.T.Professor, Centre for Nonlinear Dynamicsand its ApplicationsUniversity College London, UKDuffing equationStability
Tien, Ti.Professor, Membrane Biophysics LabMichigan State University, USABilayer lipid membranes
Tobias, DouglasAssociate Professor, Department of ChemistryUniversity of California at Irvine, USAMolecular dynamics
Toda, MorikazuEmeritus Professor, Tokyo University of EducationJapanNonlinear toys
Trueba, Jose L.ESCET, Universidad Rey Juan Carlos, Madrid, SpainBall lightning
Tsimring, Lev S.Research PhysicistSan Diego Institute for Nonlinear ScienceUniversity of California, USAAvalanches
Tsinober, ArkadyProfessor, Iby and Aladar Fleischman Facultyof Engineering, Tel Aviv University, IsraelHelicity
Tsironis, Giorgos P.Department of Physics, University of Crete, GreeceBjerrum defectsExcitonsIsing modelLocal modes in molecular crystals
Tsygvintsev, AlexeiUnite de Mathematiques Pures et Appliquees Ecolenormale superieure de Lyon, FrancePoincare theorems
Tuszynski, JackDepartment of Physics, University of Alberta, CanadaCritical phenomenaDomain wallsFerromagnetism and FerroelectricityFrohlich theoryHysteresisOrder parametersRenormalization groupsScheibe aggregates
Ustinov, AlexeyPhysikalisches Institut IIIUniversity of Erlangen-Nurnberg, GermanyJosephson junction arrays
Van der Heijden, GertCentre for Nonlinear DynamicsUniversity College London, UKButterfly effectHopf bifurcation
xxix
List of Contributors
Vazquez, LuisProfessor, Universidad Complutense de Madrid, Spain.Senior Researcher and Cofounder of the Centro deAstrobiologιaCharge density waveDispersion relationsFitzHugh--Nagumo equationVirial theoremWave propagation in disordered media
Veselov, AlexanderProfessor, Department of Mathematical SciencesLoughborough University, UKHuygens’ principle
Voiculescu, Dan-VirgilProfessor, Department of MathematicsUniversity of California, Berkeley, USAFree probability theory
Voorhees, BurtonProfessor, Department of MathematicsAthabasca University, CanadaCellular automata
Wadati, M.Professor, Department of PhysicsUniversity of Tokyo, JapanQuantum inverse scattering method
Walter, GilbertProfessor EmeritusDepartment of Mathematical SciencesUniversity of Wisconsin-Milwaukee, USACompartmental models
Waymire, Edward C.Professor, Department of MathematicsOregon State University, USAMultiplicative processes
West, BruceDepartment of Physics, US Army Research Office, USABranching lawsFluctuation--dissipation theoremKicked rotor
Wilhelmsson, HansProfessor Emeritus of PhysicsChalmers University of TechnologyGoteborg, SwedenAlfven waves
Wilson, HughCentre for Vision ResearchYork University, CanadaNeuronsStereoscopic Vision and Binocular Rivalry
Winfree, Art (Adviser)Formerly, Department of Ecology andEvolutionary BiologyUniversity of Arizona, USADimensional analysis
Wojtkowski, MaciejProfessor, Department of MathematicsUniversity of Arizona, USALyapunov exponents
Yakushevich, Ludmilla (Adviser)Researcher, Institute of Cell BiophysicsRussian Academy of Sciences, RussiaDNA solitons
Young, Lai-Sang (Adviser)Professor, Courant Institute of Mathematical SciencesNew York University, USAAnosov and Axiom A systemsHorseshoes and hyperbolicity in dynamical systemsSinai--Ruelle--Bowen measures
Yiguang Ju.Department of Mechanical Aerospace EngineeringPrinceton University, USAFlame front
Yukalov, V.I.Professor, Bogolubov Laboratory of Theoretical PhysicsJoint Institute for Nuclear Research, Dubna, RussiaBose--Einstein condensationCoherence phenomena
Zabusky, NormanProfessor, Department of Mechanical andAerospace Engineering, Rutgers UniversityNew Jersey, USAVisiometricsVortex dynamics of fluids
Zbilut, Joseph P.Associate Professor, Department of Molecular Biophysicsand Physiology, Rush UniversityChicago, USAAlgorithmic complexity
xxx
List of Contributors
Zolotaryuk, AlexBogolyubov Institute for Theoretical Physics, UkrainePolaronsRatchets
Zhou, XinProfessor, Department of MathematicsDuke University, USA
Random matrix theory IV: analytic methodsRiemann--Hilbert Problem
Zorzano, Mar�a-PazYoung Researcher, Centro de AstrobiologιaMadrid, SpainFitzHugh--Nagumo equation
xxxi
List of Entries
Ablowitz--Kaup--Newell--Segur (AKNS)system 000
Adiabatic invariants 000Alfven waves 000Algorithmic complexity 000Anderson localization 000Anosov and Axiom A systems 000Arnold diffusion 000Artificial intelligence 000Artificial life 000Atmospheric and ocean sciences 000Attractor neural network (ANN) 000Attractors 000Aubry--Mather theory 000Avalanche breakdown 000Avalanches 000Averaging methods 000Backlund transformations 000Ball lightning 000Belousov--Zhabotinsky reaction 000Bernoulli’s equation 000Berry’s phase 000Bethe ansatz 000Bifurcations 000Bilayer lipid membranes 000Billiards 000Binding energy 000Biological evolution 000Biomolecular solitons 000Bjerrum defects 000Black holes 000Born--Infeld equations 000Bose--Einstein condensation 000Boundary layers 000Boundary value problems 000Branching laws 000
Breathers 000Brownian motion 000Brusselator 000Burgers’ equation 000Butterfly effect 000Candle 000Cardiac arrhythmias and electrocardiogram 000Cardiac muscle models 000Cat map 000Catalytic hypercycle 000Catastrophe theory 000Causality 000Celestial mechanics 000Cell assemblies 000Cellular automata 000Cellular nonlinear networks 000Center manifold reduction 000Chaos vs. turbulence 000Chaotic advection 000Chaotic dynamics 000Characteristics 000Charge density wave 000Chemical kinetics 000Cherenkov radiation 000Chua’s circuit 000Clear air turbulence 000Cluster-cluster coagulation 000Coherence phenomena 000Collective coordinates 000Collisions 000Color centers 000Commensurate-incommensurate transition 000Compartmental models 000Complex Ginzburg-Landau (CGL) equation 000Conley index 000Constants of motion and conservation laws 000
xxxiii
List of Entries
Continuum approximations 000Contour dynamics 000Controlling chaos Cosmological models 000Coupled map lattice 000Coupled oscillators 000Coupled systems of partial differentialequations 000
Critical phenomena 000Damped driven anharmonic oscillator 000Darboux transformation 000Davydov soliton 000Delay-differential equations 000Denjoy theory 000Derrick--Hobart theorem 000Detailed balance 000Determinism 000Deterministic walks in random environments 000Development of singularities 000Differential geometry 000Diffusion 000Dimensional analysis 000Dimensions 000Diodes 000Discrete breathers 000Discrete nonlinear Schrodinger equations 000Discrete self-trapping system 000Dislocations in crystals 000Dispersion management 000Dispersion relations 000Distributed oscillators 000DNA premelting 000DNA solitons 000Domain walls 000Doppler shift 000Dressing method 000Dripping faucet 000Drude model 000Duffing equation 000Dune formation 000Dynamical systems 000Dynamos, homogeneous 000Economic system dynamics 000Effective mass 000Einstein equations 000Electroencephalogram at large scales 000Electroencephalogram at mesoscopic scales 000Electron beam microwave devices 000Elliptic functions 000Embedding methods 000Emergence 000Energy analysis 000Entropy 000Ephaptic coupling 000Epidemiology 000Equations, nonlinear 000
Equilibrium 000Equipartion of energy 000Ergodic theory 000Euler--Lagrange equations 000Evans function 000Evaporation wave 000Excitability 000Excitons 000Explosions 000Extremum principles 000Fairy rings of mushrooms 000Feedback 000Fermi acceleration and Fermi map 000Fermi--Pasta--Ulam (FPU) oscillator chain 000Ferromagnetism and ferroelectricity 000Fibonacci series 000Filamentation 000Fitness landscape 000FitzHugh--Nagumo equation 000Flame front 000Flip-flop circuit 000Fluctuation-dissipation theorem 000Fluid dynamics 000Fokker--Planck equation 000Forecasting 000Forest fires 000Fractals 000Framed space curves 000Franck--Condon factor 000Fredholm theorem 000Free energy 000Free probability theory 000Frenkel--Kontorova model 000Frequency doubling 000Frohlich theory 000Frustration 000Function spaces 000Functional analysis 000Galaxies 000Game of life 000Game theory 000Gel’fand--Levitan theory 000General circulation models of the atmosphere 000General relativity 000Generalized functions 000Geometrical optics, nonlinear 000Geomorphology and tectonics 000Gestalt phenomena 000Glacial flow 000Global warming 000Gradient system 000Granular materials 000Gravitational waves 000Group velocity Growth patterns 000Henon map 000
xxxiv
List of Entries
Henon--Heiles system 000Hamiltonian systems 000Harmonic generation 000Hartree approximation 000Heat conduction 000Hele--Shaw cell 000Helicity 000Hierarchies of nonlinear systems 000Higgs boson 000Hirota’s method 000Hodgkin--Huxley equations 000Hodograph transform 000Hole burning 000Holons 000Hopf bifurcation 000Horseshoes and hyperbolicity in dynamicalsystems 000
Hurricanes and tornadoes 000Huygens’ principle 000Hydrogen bond 000Hydrothermal waves 000Hysteresis 000Idempotent analysis 000Incoherent solitons 000Inertial manifolds 000Information theory 000Inhibition 000Instantons 000Integrability 000Integrable cellular automata 000Integrable lattices 000Integral transforms 000Integrate and fire neuron 000Intermittency 000Invariant manifolds and sets 000Inverse problems 000Inverse scattering method or transform 000Ising model 000Josephson junction arrays 000Josephson junctions 000Jump phenomena 000Jupiter’s Great Red Spot 000Kadomtsev--Petviashvili equation 000Kelvin--Helmholtz instability 000Kerr effect 000Kicked rotor 000Kirchhoff’s laws 000Knot theory 000Kolmogorov cascade 000Kolmogorov--Arnold--Moser theorem 000Korteweg--de Vries equation 000Kuramoto--Sivashinsky equation 000Laboratory models of nonlinear waves 000Lagrangian chaos 000Landau--Lifshitz (LL) equation 000
Langmuir--Blodgett films 000Lasers 000Lattice gas methods 000Levy flights 000Lie algebras and Lie groups 000Lifetime 000Linearization 000Liquid crystals 000Local modes in molecular crystals 000Local modes in molecules 000Long Josephson junctions 000Lorentz gas 000Lorenz equations 000Lyapunov exponents 000Magnetohydrodynamics 000Manley--Rowe relations 000Maps 000Maps in the complex plane 000Markin--Chizmadzhev model 000Markov partitions 000Martingales 000Matter, nonlinear theory of 000Maxwell--Bloch equations 000McCulloch--Pitts network 000Measures 000Mechanics of solids 000Melnikov method 000Mixing 000Modulated waves 000Molecular dynamics 000Monodromy preserving deformations 000Monte Carlo methods 000Morphogenesis, biological 000Multidimensional solitons 000Multifractal analysis 000Multiple scale analysis 000Multiplex neuron 000Multiplicative processes 000Multisoliton perturbation theory 000Myelinated nerves 000Navier--Stokes equation 000N -body problem 000Nephron dynamics 000Nerve impulses 000Neural network models 000Neuristor 000Neurons 000Newton’s laws of motion 000Nonequilibrium statistical mechanics 000Nonlinear acoustics 000Nonlinear electronics 000Nonlinear optics 000Nonlinear plasma waves 000Nonlinear Schrodinger equations 000Nonlinear signal processing 000
xxxv
List of Entries
Nonlinear toys 000Nonlinearity, definition of 000Nontwist maps 000Normal forms theory 000N -soliton formulas 000Numerical methods 000N -wave interactions 000One-dimensional maps 000Optical fiber communications 000Order from chaos 000Order parameters 000Ordinary differential equations,nonlinear 000
Overtones 000Painleve analysis 000Parametric amplification 000Partial differential equations, nonlinear 000Particle accelerators 000Particles and antiparticles 000Pattern formation 000Peierls barrier 000Pendulum 000Perceptron 000Percolation theory 000Period doubling 000Periodic bursting 000Periodic orbit theory 000Periodic spectral theory 000Perturbation theory 000Phase dynamics 000Phase plane 000Phase space 000Phase space diffusion and correlations 000Phase transitions 000Photonic crystals 000Plasma soliton experiments 000Plume dynamics 000Poincare theorems 000Poisson brackets 000Polaritons 000Polarons 000Polymerization 000Population dynamics 000Power balance 000Protein dynamics 000Protein structure 000Pump-probe measurements 000Quantum chaos 000Quantum field theory 000Quantum groups 000Quantum inverse scattering method 000Quantum nonlinearity 000Quantum theory 000Quasilinear analysis 000Quasiperiodicity 000
Random matrix theory I: origins and physicalapplications 000
Random matrix theory II: algebraicdevelopments 000
Random matrix theory III: combinatorics 000Random matrix theory IV: analytic methods 000Random walks 000Ratchets 000Rayleigh and Raman scattering and IR absorption 000Rayleigh--Taylor instability 000Reaction-diffusion systems 000Recurrence 000Regular and chaotic dynamics in atomic physics 000Relaxation oscillators 000Renormalization groups 000Rheology 000Riccati equations 000Riemann--Hilbert problem 000Rossler systems 000Rotating rigid bodies 000Rotating wave approximation 000Routes to chaos 000Salerno equation 000Sandpile model 000Scheibe aggregates 000Scroll waves 000Semiconductor laser 000Semiconductor oscillators 000Separation of variables 000Shear flow 000Shock waves 000Sinai--Ruelle--Bowen measures 000Sine-Gordon (SG) equation 000Singularity theory 000Skyrmions 000Solar system 000Solitons 000Solitons, a brief history 000Solitons, types of 000Spatiotemporal chaos 000Spectral analysis 000Spin systems 000Spiral waves 000Stability 000Standard map 000State diagrams 000Stereoscopic Vision and Binocular Rivalry 000Stochastic analyses of neural systems 000Stochastic processes 000String theory 000Structural complexity 000Superconducting quantum interference device(SQUID) 000
Superconductivity 000Superfluidity 000
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List of Entries
Superlattices 000Surface waves 000Symbolic dynamics 000Symmetry groups 000Symmetry: equations vs. solutions 000Symplectic maps 000Synchronization 000Synergetics 000Tachyons and superluminal motion 000Tacoma Narrows Bridge collapse 000Taylor--Couette flow 000Tensors 000Tessellation 000Thermal convection 000Thermo-diffusion effects 000Theta functions 000Threshold phenomena 000Time series analysis 000Toda lattice 000Topological defects 000Topology 000Traffic flow 000
Turbulence 000Turbulence, ideal 000Turing patterns 000Twistor theory 000Universality 000Van der Pol equation 000Virial theorem 000Visiometrics 000Volterra series and operators 000Vortex dynamics in excitable media 000Vortex dynamics of fluids 000Water waves 000Wave of translation 000Wave packets, linear and nonlinear 000Wave propagation in disordered media 000Wave stability and instability 000Wavelets 000Winding numbers 000Yang--Mills theory 000Zeldovich--Frank-Kamenetsky equation 000Zero-dispersion limits 000
xxxvii
Thematic list of entries
HISTORY OF NONLINEAR SCIENCE
Butterfly effect, Candle, Integrability, Celestial me-chanics, Davydov soliton, Determinism, Feedback,Fermi--Pasta--Ulam oscillator chain, Fibonacci se-ries, Hodgkin--Huxley equations, Lorenz equations,Manley--Rowe relations, Markin--Chizmadzhevmodel, Martingales, Matter, nonlinear theory of,Poincare theorems, Preface, Solar system, Soli-ton, a brief history, Tacoma Narrows Bridge col-lapse, Van der Pol equation, Zeldovich--Frank-Kamenetsky equation
COMMON EXAMPLES OFNONLINEAR PHENOMENA
Avalanches, Ball lightning, Brownian motion, But-terfly effect, Candle, Clear air turbulence, Drippingfaucet, Dune formation, Explosions, Fairy rings ofmushrooms, Filamentation, Flame front, Fluid dy-namics, Forest fires, Glacial flow, Global warming,Hurricanes and tornadoes, Jupiter’s Great Red Spot,Nonlinear toys, Order from chaos, Pendulum, Phasetransitions, Plume dynamics, Solar system, TacomaNarrows Bridge collapse, Traffic flow, Waterwaves
ANALYTICAL METHODS
Backlund transformations, Bethe ansatz, Center-manifold reduction, Characteristics, Collective coor-dinates, Continuum approximations, Dimensional
analysis, Dispersion relations, Dressing method,Elliptic functions, Energy analysis, Evans function,Fredholm theorem, Gel’fand--Levitan theory, Gen-eralized functions, Hamiltonian systems, Hirota’smethod, Hodograph transform, Idempotent anal-ysis, Inverse scattering method, Integrable trans-forms, Kirchhoff laws, Multiple scale analysis, Mul-tisoliton perturbation theory, Non-equilibrium sta-tistical mechanics, Normal forms theory, N -solitonformulas, Painleveanalysis, Periodic spectral the-ory, Perturbation theory, Phase dynamics, Poissonbrackets, Power balance, Quantum inverse scat-tering method, Quasilinear analysis, Riccati equa-tions, Rotating wave approximation, Separationof variables, Spectral analysis, Stability, State dia-grams, Synergetics, Tensors, Theta functions, Timeseries analysis, Volterra series, Zero-dispersionlimits
COMPUTATIONAL METHODS
Averaging methods, Cellular automata, Cellularnonlinear networks, Characteristics, Compartmen-tal models, Contour dynamics, Embedding meth-ods, Extremum principles, Fitness landscape, Fore-casting, Framed space curves, Hartree approxima-tion, Integrability, Inverse problems, Lattice gasmethods, Linearization, Maps, Martingales, Monte--Carlo methods, Numerical methods, Recurrence,Theta functions, Time series analysis, Visiometrics,Volterra series, Wavelets
xxxix
Thematic List of Entries
TOPOLOGICAL METHODS
Conley index, Cat map, Darboux transformation,Denjoy theory, Derrick--Hobart theorem, Differ-ential geometry, Extremum principles, Functionalanalysis, Horseshoes and hyperbolicity, Huygens’principle, Inertial manifolds, Invariant manifoldsand sets, Knot theory, Kolmogorov--Arnold--Mosertheory, Lie algebras and Lie groups, Maps, Mea-sures, Monodromy-preserving deformations, Mul-tifractal analysis, Nontwist maps, One-dimensionalmaps, Periodic orbit theory, Phase space, Renorma-lization groups, Riemann--Hilbert problem, Singu-larity theory, Symbolic dynamics, Symmetry groups,Topology, Virial theorem, Winding numbers
CHAOS, NOISE AND TURBULENCE
Attractors, Aubry--Mather theory, Butterfly effect,Chaos vs. turbulence, Chaotic advection, Chaoticdynamics, Clear air turbulence, Dimensions, En-tropy, Ergodic theory, Fluctuation-dissipation theo-rem, Fokker--Planck equation, Free probability the-ory, Frustration, Hele--Shaw cell, Horseshoes andhyperbolicity, Lagrangian chaos, Levy flights, Lya-punov exponents, Martingales, Melnikov method,Order from chaos, Percolation theory, Phase spaceanalysis, Quantum chaos, Random matrix theory,Random walks, Routes to chaos, Spatiotemporalchaos, Stochastic processes, Turbulence, Turbu-lence, ideal
COHERENT STRUCTURES
Biomolecular solitons, Black holes, Breathers, Cellassemblies, Davydov soliton, Discrete breathers,Dislocations in crystals, DNA solitons, Domainwalls, Dune formation, Emergence, Fairy rings ofmushrooms, Flame front, Higgs boson, Holons,Hurricanes and tornadoes, Instantons, Jupiter’sGreat Red Spot, Local modes in molecular crys-tals, Local modes in molecules, Multidimensionalsolitons, Nerve impulses, Polariton, Polaron, Shockwaves, Skyrmions, Solitons, types of, Spiral waves,Tachyons and superluminal motion, Turing pat-terns, Wave of translation
DYNAMICAL SYSTEMS
Anosov/axiom A system, Arnold diffusion, At-tractors, Aubry--Mather theory, Bifurcations, Bil-
liards, Butterfly effect, Catastrophe theory, Catmap, Center-manifold reduction, Chaotic dynam-ics, Coupled map lattice, Deterministic walks inrandom environment, Development of singular-ities, Dynamical systems, Equilibrium, Ergodictheory, Fitness landscape, Framed space curves,Function spaces, Gradient system, Hamiltoniansystems, Henon map, Hopf bifurcation, Horse-shoes and hyperbolicity, Inertial manifolds, In-termittency, Kicked rotor, Kolmogorov--Arnold--Moser theory, Lyapunov exponents, Maps, Mea-sures, Melnikov method, One-dimensional maps,Pattern formation, Periodic orbit theory, Phaseplane, Phase space, Phase space diffusion and cor-relations, Poincaretheorems, Reaction diffusion sys-tems, Rossler systems, Rotating rigid bodies, Routesto chaos, Sinai--Ruelle--Bowen measure, Standardmap, Stochastic processes, Symbolic dynamics,Synergetics, Universality, Visiometrics, Windingnumbers
GENERAL PHENOMENA
Adiabatic invariants, Algorithmic complexity, An-derson loccalization, Arnold diffusion, Attractors,Berry’s phase, Bifurcations, Binding energy, Bound-ary layers, Branching laws, Breathers, Brownianmotion, Butterfly effect, Causality, Chaotic dy-namics, Characteristics, Clusterr coagulation, Co-herence phenomena, Collisions, Critical phenom-ena, Detailed balance, Determinism, Diffusion, Do-main walls, Doppler shift, Effective mass, Emer-gence, Entropy, Equilibria, Equipartition of energy,Excitability, Explosions, Feedback, Filamentation,Fractals, Free energy, Frequency doubling, Frus-tration, Gestalt phenomena, Group velocity, Har-monic generation, Helicity, Hopf bifurcation, Hys-teresis, Incoherent solitons, Inhibition, Integrabil-ity, Intermittency, Jump phenomena, Kolmogorovcascade, Levy flights, Lifetime, Mixing, Modu-lated waves, Multiplicative processes, Nonlinear-ity, definition of, N -wave interactions, Order fromchaos, Order parameters, Overtones, Pattern for-mation, Period doubling, Periodic bursting, Powerbalance, Quantum chaos, Quantum nonlinearity,Quasiperiodicity, Recurrence, Routes to chaos,Scroll waves, Shear flow, Solitons, Spiral waves,Structural complexity, Symmetry: equations vs. so-lutions, Synergetics, Tessellation, Thermal convec-
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Thematic List of Entries
tion, Threshold phenomena, Turbulence, Universal-ity, Wave packets, linear and nonlinear, Wave prop-agation in disordered media, Wave stability andinstability
MAPS
Aubry--Mather theory, Backlund transforms, Catmap, Coupled map lattice, Darboux transformation,Denjoy theory, Embedding methods, Fermi acceler-ation and Fermi map, Henon map, Maps, Maps inthe complex plane, Monodromy-preserving defor-mations, Nontwist maps, One-dimensional maps,Periodic orbit theory, Recurrence, Renormaliza-tion groups, Singularity theory, Standard map,Symplectic maps
MATHEMATICAL MODELS
Ablowitz--Kaup--Newell--Segur system, Attractorneural networks, Billiards, Boundary-value prob-lems, Brusselator, Burgers equation, Cat map, Cel-lular automata, Compartmental models, ComplexGinzburg--Landau equation, Continuum approxi-mations, Coupled map lattice, Coupled systemsof partial differential equations, Delay-differentialequations, Difference equations, Discrete nonlinearSchrodinger equations, Discrete self-trapping sys-tem, Duffing equation, Equations, nonlinear, Euler--Lagrange equations, Fitzhugh--Nagumo equa-tion, Fokker--Planck equation, Frenkel--Kontorovamodel, Game of life, General circulation modelsof the atmosphere, Henon--Heiles system, Inte-grable cellular automata, Integrable lattices, Isingmodel, Kadomtsev--Petviashvili equation, Knottheory, Korteweg--de Vries equation, Kuramoto--Sivashinsky equation, Landau--Lifshitz equation,Lattice gas methods, Lie algebras and Lie groups,Lorenz model, Markov partitions, Martingales,Maxwell--Bloch system, McCulloch--Pitts network,Navier--Stokes equation, Neural network models,Newton’s laws of motion, Nonlinear Schrodingerequations, One-dimensional maps, Ordinary differ-ential equations, nonlinear, Partial differential equa-tions, nonlinear, Random walks, Riccati equations,Salerno equation, Sandpile model, Sine-Gordonequation, Spin systems, Stochastic processes, Sym-bolic dynamics, Synergetics, Toda lattice, Vander Pol equation, Zeldovich--Frank-Kamenetskyequation
STABILITY
Attractors, Bifurcations, Butterfly effect, Catastro-phe theory, Controlling chaos, Development of sin-gularities, Dispersion management, Dispersion re-lations, Emergence, Equilibrium, Excitability, Feed-back, Growth patterns, Hopf bifurcation, Lyapunovexponents, Nonequilibrium statistical mechanics,Stability
ASTRONOMY AND ASTROPHYSICS
Alfv�enwaves, Black holes, Celestialmechanics, Cos-mological models, Einstein equations, Galactic dy-namics, Gravitational waves, Henon--Heiles system,Jupiter’s Great Red Spot, N -body problem, Solarsystem
BIOLOGY
Artificial life, Bilayer lipid membranes, Biologicalevolution, Biomolecular solitons, Cardiac musclemodels, Catalytic hypercycle, Compartmental mod-els, Davydov soliton, DNA premelting, DNA soli-tons, Cardiac arrhythmias and electrocardiogram,Cardiac muscle models, Epidemiology, Excitabil-ity, Fairy rings of mushrooms, Fibonacci series,Fitness landscape, Frohlich theory, Game of life,Growth patterns, Holons, Morphogenesis, Biologi-cal, Nephron dynamics, Protein dynamics, Proteinstructure, Scroll waves, Turing patterns, Synerget-ics
CHEMISTRY
Belousov--Zhabotinsky reaction, Biomolecular soli-tons, Brusselator, Candle, Catalytic hypercycle,Chemical kinetics, Flame front, Franck--Condonfactor, Hydrogen bond, Langmuir--Blodgett films,Molecular dynamics, Oregonator, Polymerization,Protein structure, Reaction--diffusion systems, Scheibeaggregates, Vortex dynamics in excitable media
CONDENSED MATTER ANDSOLID-STATE PHYSICS
Anderson localization, Avalanche breakdown, Bjer-rum defects, Bose--Einstein condensation, Chargedensity wave, Cherenkov radiation, Color cen-ters, Commensurate--incommensurate transition,Discrete breathers, Dislocations in crystals, Domain
xli
Thematic List of Entries
walls, Drude model, Effective mass, Excitons, Fer-roelectricity and ferromagnetism, Franck--Condonfactor, Frenkel--Kontorova model, Frustration, Heatconduction,Hydrogenbond, Isingmodel, Langmuir--Blodgett films, Liquid crystals, Local modes inmolecular crystals, Mechanics of solids, Nonlin-ear acoustics, Peierls barrier, Percolation theory,Regular and chaotic dynamics in atomic physics,Scheibe aggregates, Semiconductor oscillators, Spinsystems, Superconductivity, Superfluidity
EARTH SCIENCE
Alfv�en waves, Angle of repose, Atmospheric andocean sciences, Avalanches, Ball lightning, Butterflyeffect, Clear air turbulence, Dune formation, Fairyrings of mushrooms, Forest fires, General circula-tionmodels of the atmosphere, Geomorphology andtectonics, Glacial flow, Global warming, Hurricanesand tornadoes, Kelvin--Helmholtz instability, Sand-pile model, Water waves
ENGINEERING
Artificial intelligence, Cellular automata, Cellu-lar nonlinear networks, Chua’s circuit, Controllingchaos, Coupled oscillators, Diodes, Dispersionman-agement, Dynamos, homogeneous, Electron beamdevices, Explosions, Feedback, Flip-flop circuit,Frequency doubling, Information theory, Hystere-sis, Josephson junction arrays, Josephson junctions,Langmuir--Blodgett films, Lasers, Long Joseph-son junction, Manley--Rowe relations, Neuristor,Nonlinear electronics, Nonlinear optics, Nonlinearsignal processing, Optical fiber communications,Parametric amplification, Particle accelerators, Ratch-ets, Relaxation oscillators, Semiconductor laser,Semiconductor oscillators, Superconducting quan-tum interference device, Synchronization, TacomaNarrows Bridge collapse
FLUIDS
Alfv�en waves, Atmospheric and ocean sciences,Bernoulli’s equation, Clear-air turbulence, Con-tour dynamics, Electron beam devices, Evapora-tion wave, Fluid dynamics, Forecasting, General cir-culation models of the atmosphere, Glacial flow,Hurricanes and tornadoes, Hydrothermal waves,Jump phenomena, Jupiter’s Great Red Spot, Kelvin--
Helmholtz instability, Laboratory models of non-linear waves, Lattice gas methods, Liquid crys-tals, Lorentz gas, Magnetohydrodynamics, Navier--Stokes equation, Nonlinear plasma waves, Plasmasoliton experiments, Plume dynamics, Rayleigh--Taylor instability, Shear flow, Shock waves, Super-fluidity, Surface waves, Taylor--Couette flow, Ther-mal convection, Thermo-diffusion effects, Trafficflow, Turbulence, Turbulence, ideal, Visiometrics,Vortex dominated flows, Water waves
NEUROSCIENCE
Artificial intelligence, Attractor neural network, Cellassemblies, Compartmental models, Depth percep-tion and binocular rivalry, Electroencephalogram atlarge scales, Electroencephalogram at mesoscopicscales, Ephaptic coupling, Evans function, Gestaltphenomena, Hodgkin--Huxley equations, Integrateand fire neuron, Multiplex neuron, Myelinatednerves, Nerve impulses, Neural network models,Neurons, Pattern formation, Perceptron, Stochasticanalyses of neural systems, Synergetics
NONLINEAR OPTICS
Damped driven anharmonic oscillator, Cherenkovradiation, Color centers, Dispersion management,Distributed oscillators, Excitons, Filamentation, Ge-omentrical optics, nonlinear, Harmonic generation,Hole burning, Kerr effect, Lasers, Liquid crystals,Maxwell--Bloch system, Nonlinear optics, Opticalfiber communications, Photonic crystals, Polariton,Polaron, Pump-probe measurements, Rayleigh andRaman scattering and IR absorption, Semiconductorlaser, Tachyons and superluminal motion
PLASMA PHYSICS
Alfv�en waves, Ball lightning, Charge-density wave,Drude model, Dynamos, homogeneous, Electronbeam devices, Magnetohydrodynamics, Nonlinearplasma waves, Particle accelerators, Plasma solitonexperiments
SOCIAL SCIENCE
Economic dynamics, Epidemiology, Game theory,Hierarchies of nonlinear systems, Population dy-namics, Traffic flow
xlii
Thematic List of Entries
SOLID MECHANICS ANDNONLINEAR VIBRATIONS
Angle of repose, Avalanche breakdown, Bilayerlipid membranes, Bjerrum defects, Charge-densitywave, Cluster coagulation, Color centers, Detailedbalance, Dislocations in crystals, Domain walls,Frustration, Glacial flow, Granular media, Growthpatterns, Heat conduction, Hydrogen bond, Isingmodel, Kerr effect, Langmuir--Blodgett films, Liquidcrystals, Localmodes inmolecular crystals,Mechan-ics of solids, Molecular dynamics, Nonlinear acous-tics, Protein dynamics, Ratchets, Rheology, Sandpilemodel, Scheibe aggregates, Shock waves, Spin sys-tems, Superlattices, Tessellation, Topological defects
THEORETICAL PHYSICS
Berry’s phase, Black holes, Born--Infeld equation,Celestial mechanics, Cherenkov radiation, Coner-
vationlaws and constants of motion, Cosmologicalmodels, Critical phenomena, Derrick--Hobart theo-rem, Detailed balance, Einstein equations, Entropy,Equipartition of energy, Fluctuation-dissipation the-orem, Fokker--Planck equation, Free energy, Galax-ies, General relativity, Gravitational waves, Hamil-tonian systems, Higgs boson, Instantons, N-bodyproblem, Newton’s laws of motion, Nonlinear the-ory of matter, Particles and antiparticles, Quantumfield theory, Quantum theory, Regular and chaoticdynamics in atomic physics, Rotating rigid bodies,Skyrmions, String theory, Tachyons and superlumi-nal motion, Twistor theory, Virial theorem, Yang--Mills theory
xliii