n-dimensional dynamical systems exploiting instabilities ... · n-dimensional dynamical systems...

N-dimensional dynamical systems exploiting instabilities in fullJ. Rius, M. Figueras, R. Herrero, J. Farjas, F. Pi et al. Citation: Chaos 10, 760 (2000); doi: 10.1063/1.1324650 View online: http://dx.doi.org/10.1063/1.1324650 View Table of Contents: http://chaos.aip.org/resource/1/CHAOEH/v10/i4 Published by the American Institute of Physics. Additional information on ChaosJournal Homepage: http://chaos.aip.org/ Journal Information: http://chaos.aip.org/about/about_the_journal Top downloads: http://chaos.aip.org/features/most_downloaded Information for Authors: http://chaos.aip.org/authors

Downloaded 30 Apr 2013 to 84.88.138.106. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://chaos.aip.org/about/rights_and_permissions

http://chaos.aip.org/?ver=pdfcov

http://aipadvances.aip.org?ver=pdfcov

http://chaos.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=J. Rius&possible1zone=author&alias=&displayid=AIP&ver=pdfcov

http://chaos.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=M. Figueras&possible1zone=author&alias=&displayid=AIP&ver=pdfcov

http://chaos.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=R. Herrero&possible1zone=author&alias=&displayid=AIP&ver=pdfcov

http://chaos.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=J. Farjas&possible1zone=author&alias=&displayid=AIP&ver=pdfcov

http://chaos.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=F. Pi&possible1zone=author&alias=&displayid=AIP&ver=pdfcov


http://link.aip.org/link/doi/10.1063/1.1324650?ver=pdfcov

http://chaos.aip.org/resource/1/CHAOEH/v10/i4?ver=pdfcov

http://www.aip.org/?ver=pdfcov


http://chaos.aip.org/about/about_the_journal?ver=pdfcov

http://chaos.aip.org/features/most_downloaded?ver=pdfcov

http://chaos.aip.org/authors?ver=pdfcov

CHAOS VOLUME 10, NUMBER 4 DECEMBER 2000

D

N-dimensional dynamical systems exploiting instabilities in fullJ. Rius and M. FiguerasDepartament de Fı´sica, Universitat Auto`noma de Barcelona, 08193 Bellaterra, Spain

R. HerreroDepartament de Meca`nica de Fluids, Termote`cnia i Fısica, EUETIB, Universitat Polite`cnia de Catalunya,Comte d’Urgell, 187, 08036 Barcelona, Spain

J. FarjasDepartament de Fı´sica, Universitat de Girona, Avinguda Santalo´, s/n, 17071 Girona, Spain

F. Pi and G. OrriolsDepartament de Fı´sica, Universitat Auto`noma de Barcelona, 08193 Bellaterra, Spain

~Received 31 May 2000; accepted for publication 21 September 2000!

We report experimental and numerical results showing how certainN-dimensional dynamicalsystems are able to exhibit complex time evolutions based on the nonlinear combination ofN-1oscillation modes. The experiments have been done with a family of thermo-optical systems ofeffective dynamical dimension varying from 1 to 6. The corresponding mathematical model is anN-dimensional vector field based on a scalar-valued nonlinear function of a single variable that is alinear combination of all the dynamic variables. We show how the complex evolutions appearassociated with the occurrence of successive Hopf bifurcations in a saddle-node pair of fixed pointsup to exhaust their instability capabilities inN dimensions. For this reason the observedphenomenon is denoted as the full instability behavior of the dynamical system. The processthrough which the attractor responsible for the observed time evolution is formed may be rathercomplex and difficult to characterize. Nevertheless, the well-organized structure of the time signalssuggests some generic mechanism of nonlinear mode mixing that we associate with the cluster ofinvariant sets emerging from the pair of fixed points and with the influence of the neighboringsaddle sets on the flow nearby the attractor. The generation of invariant tori is likely during the fullinstability development and the global process may be considered as a generalized Landau scenariofor the emergence of irregular and complex behavior through the nonlinear superposition ofoscillatory motions. ©2000 American Institute of Physics.@S1054-1500~00!01004-1#

-

-

,

f

s

s

inthetice-beg-ld.

calin-

Oscillatory phenomena are ubiquitous in natural and so-cial systems and their investigation is currently donewithin the context of nonlinear dynamics. The oscillationsin a given system may appear associated either with intrinsically sustained mechanisms or with externallymodulated inputs. Processes with different time scales often coexist and in certain cases the interrelation of oscil-lations produces complex evolutions in which, howevercyclic repetitions are usually apparent. For instance, inbiology, a direct example of this behavior is found in thebursting response of small neural networks in the stoma-togastric nervous system of crustacea,1 while a more in-volved example could be the wake–sleep cycle of a brain.Leaving apart the case of external modulations, we find itinteresting to understand how a nonlinear system canproduce different characteristic frequencies and how itcan mix the corresponding oscillations to yield complextime evolutions. Although the problem looks basic andsimple its answer is pending. The paradigm of chaos isnot useful here and among the known mechanisms ononlinear dynamics only the Landau scenario can be in-voked. The present work is a contribution to the enlight-enment of this problem. We report experimental and nu-merical results showing the emergence of complexity

7601054-1500/2000/10(4)/760/11/$17.00

ownloaded 30 Apr 2013 to 84.88.138.106. This article is copyrighted as indicated in the abstract. R

through the nonlinear superposition of oscillatory mo-tions in a dynamical system. The phenomenon developin a generalized Landau scenario where the oscillationsappear in association with the Hopf bifurcations of a setof fixed points and the complexity arises from „i… thenumber of different characteristic frequencies, and „ii …the variety of forms through which the nonlinear mecha-nisms combine the oscillation modes. The phenomenon irelevant because it illustrates how the nonlinear modemixing works in nonlinear dynamics and it would prob-ably be involved in any system exhibiting various self-sustained oscillations simultaneously.

I. INTRODUCTION

Complexity may emerge through a variety of waysnonlinear dynamics although it is mostly associated withirregularity of chaos. The peculiar properties of the chaostates2 are compatible with a low number of degrees of fredom, while additional levels of complex dynamics canintroduced in high-dimensional systems by properly aumenting the structure of the nonlinear part of the vector fieFor instance, the effective participation of more dynamivariables within the nonlinear feedback may enhance the

© 2000 American Institute of Physics

euse of AIP content is subject to the terms at: http://chaos.aip.org/about/rights_and_permissions

tsnn

etc

r-acth

eiteau-il

ci

d

toremle

tinic

o-ldth

eie

ean

hc

ngitblennane

lab

dtyf

iain

ales

n-

os-

torirca-nningun-ex-theicheli-ngde-on

hex

ehav-in allewedi-

eth-re-

pto-

oth

ao-ed

ase-

ex

s-o-

tureer

eteroto-

ser

ed

761Chaos, Vol. 10, No. 4, 2000 N-dimensional dynamical systems

D

stability capabilities of the fixed points and other limit seThis article considers a situation of this type and preseexperimental and numerical results showing the emergeof complex time dynamics independently of chaos.

The problem we are dealing with is how a large numbof characteristic frequencies can successively appear insystem response and how the corresponding oscillationsmix to yield complex time evolutions. The theory of bifucations shows that the exclusive way for introducing charteristic frequencies into the time dynamics is throughvariety of Hopf-type two-dimensional instabilities,2 i.e., thePoincare´-Andronov–Hopf bifurcation of a fixed point, thNaimark–Sacker or secondary Hopf bifurcation of a limcycle, and the successive bifurcations originating highorder invariant tori. This relates our problem to the Landproposal3 for tentatively explaining the emergence of turblence through an indefinite sequence of oscillatory instabties that, in light of the bifurcation theory, is usually assoated with a sequence of torus bifurcations.4 The Landausequence is not considered a route to chaos because itnot produce sensitivity to initial conditions5 and this is instrong contrast with the already existing strange attracwhen triply periodic flows on three-dimensional tori aperturbed.6 On the other hand, the lack of dissipative systeexhibiting such large sequences of torus bifurcations hasthe Landau scenario as a hypothetical way for incorporaadditional degrees of freedom into the oscillatory dynamof high-dimensional systems.

This work shows that the combination of oscillatory mtions in a nonlinear dynamical system can effectively yiecomplex time evolutions evoking the Landau idea aboutemergence of irregularity. This behavior has been foundsystems able to exploit all the instability capabilities of thfixed points through Hopf bifurcations. In relation to thLandau scenario, our problem is simpler because it dwith systems of finite dimension, but it is enriched by cosidering~i! more than one fixed point and~ii ! the occurrenceof successive Hopf bifurcations on each fixed point. Tformer is relevant because the nonlinear mechanismsmix the oscillatory dynamics emerging from neighboripoints, while the latter implies a variety of coexisting limcycles and the possibility of different sequences of torusfurcations. Nevertheless, as it will be shown, the comptime evolutions observed under these circumstances cabe explained by means of the torus bifurcations alonemore general and more robust mechanisms of nonlinmode mixing must be invoked.

More concretely, we deal withN-dimensional dynamicasystems possessing a nonlinear function of a single varithat, in its turn, is a linear combination of theN dynamicalvariables. The fixed points of these systems appear alignephase space in an alternate sequence of saddle-nodeThe saddle separatrices determine the attraction basins onodes and the basic dynamical phenomena will be assocwith an attractor arising from one of the nodes and growunder the influence of the nearest saddle point. InN dimen-sions, a saddle-node pair of fixed points can sustain a totN-1 Hopf bifurcations, occurring on either one or the othpoint,7 and affecting differently oriented planes of the pha


.tsce

rhean

-e

r-u

i--

oes

rs

sftgs

einr

ls-

ean

i-xotd

ar

le

inpe.thetedg

ofre

space. The points initially have stable manifolds of dimesion N-1 andN, respectively, and after theN-1 bifurcationsone of them has become fully unstable while the other psesses only one stable dimension.N-1 limit cycles have suc-cessively emerged from the points and some invariantcould have been created through secondary Hopf bifutions of the cycles.8 The cluster of limit sets contains aattractor at least9 and a variety of saddles with the commofeature of having a branch of their unstable manifold endtoward the attractor. The secondary processes occurringder such circumstances may be rather complex, but theperimental and numerical results show that they producenonlinear mixing of oscillation modes with relatively generfeatures. In essence, the attractor incorporates localizedcal motions related to the influence of the neighborisaddles and, in this way, the observed time dynamicsscribes an irregular succession of oscillatory trains basedtheN-1 characteristic frequencies initially generated from tpair of fixed points. We call the exhibition of such completime waveforms the full instability behavior of thN-dimensional system and a detailed analysis of such beior within a more general context has been presentedseparate paper.10 Our aim here is to demonstrate the fuinstability behavior with a family of physical devices whoseffective dynamical dimension may easily be varied anddescribe experimental results for successively increasingmensions up toN56. The interpretation is sustained with thlinear stability analysis and numerical simulations of a maematical model reproducing correctly the experimentalsults.

II. NONLINEAR SYSTEM

The nonlinear systems are based on the so-called othermal bistability with localized absorption~BOITAL !11

and they have been described in detail elsewhere from bthe experimental12 and mathematical13,14 points of view. ABOITAL device consists of a Fabry-Pe´rot cavity in whichthe input mirror is partially absorbing and the spacer ismultilayer of transparent materials with alternatively oppsite thermo-optic effects. Concretely, in this work we uslayers of glass, sunflower oil, silicone~F600, Bayer AG!,optical adhesive~NOA61, Norland!, and optical gel~0608,Cargille! with thicknesses ranging frommm to mm. Thermalexpansion works in the case of glass as a positive phshifting effect (1025 K21) while the rest of the materialsproduce negative shifting effects due to refractive indchanges~22 to 2531024 K21!. The cavity mirrors were ahigh reflection ~.0.98! dielectric multilayer and a 7 nmnickel-chrome film having reflections of about 0.2 and tranmission of about 0.4. The device was placed on a thermelectric plate to define better the environment temperaand was irradiated for the metal mirror side with a lasbeam of 514.5 nm wavelength focused to a 0.3 mm diamspot. The reflected light was detected by means of a phdiode and a signal proportional to the reflected power,PR ,was digitized and stored in a computer. The incident lapower,PE , was used as the control parameter.

The time dynamics in BOITAL devices is associat


gouesv

indihendofthiteiesive

opelae.

siea,ean

ipti

anteacul

edcoinano-onowth

dre

nd

atn

nst-or:

re-

n

hetheon

-setewm-henn

eri-e

icalde-

ecertate

fur-in

a-

762 Chaos, Vol. 10, No. 4, 2000 Rius et al.

D

with the heat propagation from the absorbing mirror throuthe cavity spacer, while the light provides an instantanenonlinear feedback to the heat source. The light wave tthe spacer temperature by means of its phase shift in a caround-trip and transfers such information to the absorbmirror by means of interference effects. The temperaturetribution remains practically unchanged during the liground-trip and determines therefore the interference statthat time. On the other hand, the light interference depenonlinearly on the phase shift through the Airy functionthe cavity and it constitutes the exclusive nonlinearity ofsystem. In addition, the multilayer of alternatively opposthermo-optic materials can originate oscillatory instabilitbecause~i! the temperature variations produce competitcontributions to the light phase shift and,~ii ! such contribu-tions are time delayed according to the relative positionthe layers with respect to the absorbing mirror. With a prochoice of materials and thicknesses, the various spacingers behave as effective degrees of freedom and the numblayers determines the dynamical dimension of the system

Some materials, like the adhesives and gels, exhibitnificant and opposite phase-shifting effects due to bothpansion and refraction, and usually these effects have redifferent time constants. Under proper circumstancessingle layer of one of such materials can introduce twofective degrees of freedom into the system dynamicshigher dimensionalities may be experimentally achievedthis way. This behavior has been known since the first ocal bistability experiments on self-sustained oscillationssemiconductors15 and more recently for the case of an opticadhesive.16 On the other hand, the phase-shifting coefficieof these materials exhibit a significant temperature depdence that can be used for a fine adjustment of the spstructure by means of the environment temperature regtion.

The physical description of a BOITAL system is bason the homogeneous heat equation subject to the propertinuity and boundary conditions, of which the one describthe localized heat source by light absorption is nonlocalnonlinear.13 The linear stability analysis of the stationary slution points out clearly an effective dynamical dimensiequal to the number of spacing layers and it has been shthat the partial differential equation may be reduced tofollowing dimensionless low-order model:14

dc j

dt52(

i 51

N

bji ~c i2aiA~c!mE!, j 51,2, . . . ,N, ~1a!

with

c5c01(1

N

c j , ~1b!

wherec is the round-trip phase shift andc0 is its value inthe absence of laser heating.N is the number of layers aneach variablec j denotes the variation due to temperatuchanges of the phase shift associated with thej th layer.c j isproportional to the space-averaged temperature acrosslayer and to the thermo-optic coefficient of the correspoing material. The parametersbji andai depend on the cavity


hstsitygs-tats

e

fry-

r of

g-x-llya

f-d

ni-nlsn-era-

n-gd

ne

the-

spacer properties and thermal boundary conditions,14 andmE

is the incident light intensity normalized in such a way th(ai51. The ratesbji describe the thermal coupling betweelayers and the associated diffusion times, whileaj character-izes the effective contribution of thej th layer to the phaseshift variations.A(c) describes the light interference withithe absorbing film and it is a positive-defined almosinusoidal function depending only on the mirrparameters.13 It may be written in the following closed form

A~c!5m1 cosc2m2

cosc2m3, ~2!

very convenient for the numerical simulations, and thesults reported in the paper correspond tom151.06, m2

51.25, andm351.86. The reflected light intensity is giveby R(c)mE , with the interferometer reflectionR(c).12A(c). Thec(t) evolution will typically present variationslarger than 2p and supplementary foldings appear then in treflected power signal. Such foldings simply describephase shift overcoming the maximum or minimum reflectivalues and lack of dynamical significance~see, e.g., Fig. 2!.

It is useful to know that the system~1! admits to beinglinearly transformed to a canonical form as follows:14

x152(j 51

N

cjxj1A~c!mE ,

~3a!x j5xj 21 , j 52, . . . ,N,

with

c5c01(j 51

N

djxj , ~3b!

where the coefficientscq anddq are functions of thebji andai and, in particular,dN5cN . For cavity spacers of alternatively opposite thermo-optic materials, the correspondingof dj values present alternatively opposite signs. The nvariables lack of direct physical interpretation, but the siplicity of the canonical form facilitates the analysis and tcomparison with other models. In addition, we have show10

that the linear stability analysis of~3! can be used to desigN-dimensional systems, i.e., to determine theircj anddj val-ues, in which the fixed points of a saddle-node pair expenceN21 Hopf bifurcations with preselected values for thoscillation frequency and control parameter. The numersimulations reported in this paper correspond to systemssigned with this method.

The steady-state solution of Eqs.~1! or ~3! as a functionof mE is determined byA(c) and consists of successivS-shaped branches. BOITAL devices with different spastructures but equal mirrors have the same steady-sbranching diagram describingc vs mE . The properties of thecavity spacer determine the power scale factor of the bication diagram through the normalization factor includedmE and, most importantly, the possibility of oscillatory instbilities on the steady-state branches.13,14


oein

et

esinwbro

mal

clanwothione

heercr

sid

eA-no

o

oc

ctorons.

chrdndteserex-

sive

r a

rs

meur-ter-ria-

asic

turesn-


D

III. DYNAMICAL PHENOMENA FOR SUCCESSIVELYINCREASING DIMENSION

The BOITAL family enables us to analyze systemssuccessively increasing dimension for the gradual undstanding of complex time evolutions. With this aim we begby briefly describing known results forN51, 2, and 3, andthen present experimental results forN54 and 6, and nu-merical simulations forN56 and 10.

Figure 1 shows the response of a BOITAL cavity spacwith a single material.11 It represents the reflected lighpower when the incident power is slowly varied with succsive back and forth sweepings. The device exhibits switchjumps and the consequent hysteresis cycles associatedpairs of saddle-node bifurcations. The saddle solution intween two stable branches plays in this case the simpleof a separatrix in a one-dimensional phase space.

The presence of a second material with opposite theroptic effect within the cavity results in proper dynamicphenomena of two-dimensional phase spaces.17 The bifurca-tion diagram is also formed by successive hysteresis cybut, in addition, it contains oscillatory states that appeardisappear by means of a Hopf bifurcation occurring ttimes on each node branch. Near the Hopf bifurcationfrequency is the same in all the branches but the oscillatmay suffer the influence of a neighboring saddle point whthe limit cycle grows. As shown in the example of Fig. 2, toscillation period strongly increases until the orbit maktangency to the saddle and vanishes in a homoclinic bifution, after which the system evolves toward the oscillatostate emerging from the node point located at the otherof the saddle separatrix.

Figure 3 shows the three basic kinds of thredimensional dynamics observed in the response of BOITcavities.12,18 The stable limit cycle born in the Hopf bifurcation of a node point passes near a saddle limit set thatmay be either a saddle focus with inward spiraling@Fig. 3~a!#or a saddle limit cycle generated by a Hopf bifurcationthat point@Fig. 3~b!#. In this way, the time evolution of theattractor incorporates the faster oscillation frequency ass

FIG. 1. Reflected light power as a function of the incident power foBOITAL cavity spaced with 200mm of sunflower oil.


fr-

d

-gith

e-le

o-

esd

es

n

sa-ye

-L

w

f

i-

ated with the saddle limit set, and the nearer the attrapasses to the saddle the larger the number of fast oscillatiIn the third kind of dynamics@Fig. 3~c!#, the stable limitcycle bends by reinjecting toward the inner point from whiit originated and which now is a saddle focus with outwaspiraling. The reinjection bending is related to a distant alarge saddle limit cycle and the fast reinjection peak denothe characteristic time of that cycle. When the incident powis increased, the attractor grows and the approach to theternal saddle cycle produces a higher number of succes

FIG. 2. Time evolution of the reflected power for different incident poweobserved in a two-layer device spaced with 140 and 75mm of glass andsunflower oil, respectively. The vertical scale in arbitrary units is the safor all the recordings. The lower signal is a transient indicating the occrence of a homoclinic bifurcation in between 35.5 and 35.6 mW. The inferometric foldings in the reflected power signals denote phase shift vations larger than 2p and have no dynamical significance.

FIG. 3. Time evolutions and reconstructed attractors showing the bkinds of three-dimensional dynamics observed in BOITAL cavities.~a! and~b! correspond to a two-layer device spaced with 140 and 540mm of glassand optical adhesive, respectively, but at different environment tempera@25 °C for ~a! and 18 °C for~b!#. The adhesive plays a twofold role resposible for the three-dimensional behavior of the two-layer device.~c! Corre-sponds to a three-layer device spaced with 140mm, 35 mm, and 1 mm ofglass, sunflower oil, and glass, respectively.


thwidblrs

es

it

th

ani-

bebinteowo

min

rbindin

am

to

n-sr-

aep-o-

st-ut

ausjustw-ive

ractorendle8.2

theadfre-ar-ot

noond

eceTh

a

yerwced


D

peaks before each outward spiraling. Thus underlyingdynamics there are homoclinic connections associatedsaddle limit sets arising from the original saddle and nofixed points and having two- and one-dimensional stamanifolds, respectively. Under clear dominance of the fikind of homoclinicity, the system evolution describShil’nikov-type attractors,12 while the other kind of homocli-nicity produces Ro¨ssler-type folded bands.18 In the parameterspace, the variety of dynamics appears organized aroundhomoclinic cycle connecting the two kinds of saddle limsets.

Figure 4 presents numerical results illustrating hownonlinear mixing works in a Shil’nikov-type attractor forN53. The initial node point has produced the stable cyclenow is a saddle focus with outgoing spiraling, while the intial saddle point has generated a saddle limit cycle bycoming fully unstable. One of the branches of the unstamanifold of the saddle limit cycle approaches the attractcycle in a well-defined place. The spiral motion associawith the stable manifold of the saddle cycle affects the flaround the unstable manifold and works like a corkscrewthe stable limit cycle when it grows under the control paraeter variation. The helical motion of the attractor evolvestime according to the oscillations of the saddle periodic oand mode mixing then occurs. The stable periodic orbitcorporates additional helical turns through a continuousformation and, during the process, it can be involvedperiod-doubling and cyclic saddle-node bifurcations.19 In

FIG. 4. Phase space representation of numerical results illustrating thelinear mixing of the oscillation modes associated with the Hopf bifurcatiof a saddle-node pair of fixed points forN53. The calculations corresponto the dimensionless physical parametersgj51,0.8,1, h j51,26.9,15.37,kj5D j51,0.1,1,hF5hB50.5, j 51,2,3, whose definition and relation to thcoefficients of Eqs.~1! are given in Ref. 14. The almost conical gray surfadescribes one side of the unstable manifold of the saddle limit cycle.stable cycle has emerged from the node point formE513.10 and has grownwithout suffering any bifurcation up tomE513.36, but it has incorporatednumber of helical turns around the saddle unstable manifold.


etheet

the

e

d

-legd

n-

it-e-

this way, a complex attractor can develop together withlarge number of nonstable periodic orbits and all of thewill grow by transforming under the corkscrew effect upbeing destroyed at the homoclinic bifurcation.

Figure 5 illustrates an experimental example of full istability behavior forN54. The sequence of time evolutionfor different incident light powers was obtained with a foulayer device of $glass-silicone-glass-sunflower oil% withthicknesses of 140, 35, 400, and 305mm, respectively. Thefast oscillations shown forPE587 mW appeared throughsupercritical Hopf bifurcation of a stable fixed point. Thslow frequency modulation of the fast oscillations also apeared supercritically and it denotes the creation of a twtorus through a secondary Hopf bifurcation of the fafrequency limit cycle. The stable torus grows with the inppower by approaching the external saddle focus andShil’nikov-type dynamics is generated. The spiraling focintroduces oscillations at the intermediate frequency and,before homoclinicity, the signal becomes aperiodic by shoing a different number of such oscillations at the successpassages near the saddle. The process ends with the attdestruction in the homoclinic bifurcation and the system thshifts to an oscillating state at the other side of the sadseparatrix, as evidenced by the transient signal for 17mW.

Figure 6 corresponds to a very similar device as incase of Fig. 5 but with a thinner layer of silicone, 30 insteof 35 mm. This system generates the same oscillationquencies but with the slower and faster oscillations appeing in the reverse order. For higher incident powers n

n-s

e

FIG. 5. Time evolutions observed in the reflected power of a four-laBOITAL device for different incident light powers. The signals show hothe two-frequency oscillation associated with an invariant torus is influenby the attracting spiral of an external saddle focus.


ato

ths

edgslesatheinFionwu-

elofudeite

oponub

erizecsled

stro

ngthet. Itrns

thetroled ineral

ofint

ss,x-c-ddle

wo

toerm

.

theenhiteandddle

yothea, theecondThe,

-


D

shown in Fig. 6, the evolution incorporates the intermedifrequency of the external saddle focus and signals almequal to those of Fig. 5 are obtained.

By considering the parameter space, it seems clearthe devices of Figs. 5 and 6 correspond to different sidea codimension-two bifurcation of type (6 iva ,6 ivb), inwhich the curves of two Hopf bifurcations of the same fixpoint cross one another. The theory of universal unfoldin2

shows that secondary Hopf bifurcations of the limit cyccan also emerge from this eigenvalue degeneracy and thfirst approximation, the two-frequency evolutions over ttorus are based on the Hopf frequencies of the fixed poThe presence of an invariant torus is clear in the case of5 but not so in the case of Fig. 6, where the fast oscillatiemerge from nothing in two well-defined places of the slofrequency limit cycle. Very similar time evolutions were nmerically obtained from models~1! or ~3! and the continuousfollowing of the low-frequency limit cycle indicates that thlocalized packets of fast oscillations emerge without anycal bifurcation. Thus a mechanism other than the torus bication must be invoked in order to explain this kind of momixing, and Fig. 7 presents numerical results illustratingWe are dealing with a four-dimensional phase space whan initially stable fixed point has done two successive Hbifurcations, and the saddle limit cycle created at the secbifurcation is now near to becoming stable by doing a scritical torus bifurcation.20 In this situation, i.e., when thetorus bifurcation will occur on the saddle limit cycle raththan on the stable one, the stable cycle exhibits a localstructure of helical motion in which the time dynamievolves according the oscillations of the saddle limit cycIn the case of Fig. 7 neither the stable orbit nor the sadorbit have experienced any local bifurcation21 and the helicalmotion ~with the associated time dynamics! has appeared aa gradual deformation of the stable orbit during the con

FIG. 6. Evolutions obtained with a four-layer device slightly differentthat of Fig. 5. In this case the slow and fast oscillations appear in revorder and the creation of an invariant torus does not seem likely. Interdiate frequency oscillations will appear also for higher incident powers


est

atof

, in

t.g.s-

-r-

.refd-

d

.le

l

parameter increase. The origin of this kind of mode miximust be related to the influence of the flow dynamics ofsaddle orbit toward a well-defined place of the stable orbiis worth remarking that a second structure of helical tualways appears on the stable orbit~in the zone denoted by ain Fig. 7! when the control parameter is increased, andtwo structures usually connect together for higher conparameter values. This double structure can be appreciatthe experimental results of Fig. 6 and it constitutes a genfeature also observed for higher dimensions~see the evolu-tion for 66.3 mW in Fig. 8!. On the other hand, in the caseFig. 7, there is no influence of the external saddle fixed pobecause it is very far from its Hopf bifurcation. Neverthelein situations of full instability behavior, the attractor can ehibit the superposition of differently oriented helical strutures associated with both the external and internal salimit sets. The time dynamics of the variablec typicallyexhibits the influence of the internal saddle sets in the t

see-

FIG. 7. Numerical results forN54 illustrating the nonlinear mode mixingof the two oscillation modes emerging in successive Hopf bifurcations ofsame fixed point, for a situation in which the torus bifurcation will happon the saddle orbit created at the second bifurcation. The black and wthick lines describe the stable and saddle periodic orbits, respectively,the white cross denotes the fixed point. The unstable manifold of the saorbit is represented by means of a number of trajectories~Ref. 24! depictedin a thin black line.~a!, ~b!, and~c! are projections in the planes defined bdifferent pairs of variables.~d! presents the time evolution of the twperiodic orbits. The unstable manifold is three-dimensional, whilestable manifold~not drawn! is two-dimensional and does not work likeseparatrix. The numbers indicate certain places on the stable orbitsame for the various representations. The label a denotes where a shelical structure will appear by increasing the control parameter.calculations were done with model~3! for cq550,438.98,480.71,358.22dq5217.601,66.044,2204.46,358.22,q from 1 to 4, andmE510. TheHopf bifurcations of the initially stable fixed point occurred formE58.2with angular frequency of 1.41, andmE59.1 with frequency of 25, respectively, and the torus bifurcation of the saddle orbit will occur formE

510.3 with the secondary frequency equal to 1.38.


the

weics.of

,togelthear-earThe

yen-e.ut. 8,atal

r-

beati-fur-dle

lu-of

ncy

dle

ec-

ableer asny

-heointe

-n-

enalthe

the

fncetsx-

s

oerfT

r thtio


D

FIG. 8. Experimental time evolutions showing the nonlinear combinationfive oscillation modes in the reflection of a BOITAL device with a five-layspacer. The system exhibits six-dimensional dynamics because one olayers introduces two degrees of freedom into the nonlinear feedback.vertical and horizontal scales are common for all the signals, except fotime expanded details. The numbers in italics denote the different oscillamodes ordered from low to high frequency.


lateral sides of the lowest-frequency undulations, whileexternal saddle sets affect the top and the bottom.

Figure 8 presents really complex time evolutions thatinterpret as corresponding to a six-dimensional dynamThe recordings were obtained with a five-layer device$glass-silicone-glass-gel-glass% with thicknesses of 140, 35400, 180, and 3000mm, respectively. The system was ableexhibit five oscillation modes supposedly because thelayer introduced a double degree of freedom throughthermal expansion and thermo-optical effects. The five chacteristic times determined when the oscillations appnearly sinusoidal are 310, 23, 2.2, 0.35, and 0.07 s.corresponding oscillation frequencies are denoted byv j ,with j from 1 to 5, and they are indicated in the figure bmeans of the numberj. Notice that, in certain cases, themergence of fast oscillations in the middle of a slower udulation may enlarge the corresponding characteristic tim

The signals for successive input light powers point ohow the waveform structures appear. In the case of Figthe oscillations begin with a supercritical Hopf bifurcationv1 on the node point, but soon incorporate two additionfrequencies,v3 andv5 , in the two lateral structures appeaing on eachv1 undulation ~see detail for 66.3 mW!. Therelation of the new frequencies with the node point cannotverified in the experiment, but the analysis of the mathemcal model shows that they emerge in successive Hopf bications of this point by means of the corresponding sadlimit cycles. The numerical simulations indicate that evotions like that for 66.3 mW appear without the occurrencetorus bifurcations on the stable cycle. The three-frequewaveform may be interpreted as thev1 stable limit cycleinfluenced by the out structures of either the pair of sadperiodic orbits or, more probably, a (v3 ,v5) saddle torusderived from one of these cycles. In other words, we conjture a situation similar to that of Fig. 7, but forN56, wherea saddle two-torus has appeared in the center of the stlimit cycle and where the two-frequency motion of thsaddle torus is transferred to certain places of the attractodefined by the approach of the unstable manifold. In acase, the oscillations atv1 , v3 , andv5 seem clearly associated with the node point and limit sets derived from it. Tapproach of the three-frequency attractor to the saddle pmanifests first through thev2 oscillations appearing at thtop of thev1 oscillations~105.8 mW!. At higher powers thev2 oscillation mixes with the (v3 ,v5) structure while somev4 oscillations also appear~134.8 mW!. The uniform ampli-tude of thev1 , v3 , andv5 oscillations indicates the occurrence of the corresponding Hopf bifurcations, while the covergence of thev2 and v4 oscillations suggests that thsaddle is even an attractive bifocus with a one-dimensiounstable manifold. The 146.6 mW signal indicates thatsaddle point has already made the Hopf bifurcation atv2 .This is clearly denoted by the sporadic passages nearbifocus point with convergence atv4 and divergence atv2

~see detail for 146.6 mW!. In addition, the large number ov2 oscillations with uniform amplitude suggests the preseof the saddle limit cycle and the relative proximity of ihomoclinic connection. Homoclinic chaos may then be epected to occur in accordance with Shil’nikov’s theorem22

f

theheen


asresr

mpsd

iesop

tio

o

an

ormthononim-hecethhaheila-te

d

thed,

pos-r-

bi-ns.s-the

02,fre-non-ilar


D

and, in fact, the waveform structure for 146.6 mW is notrepetitive as for lower light powers. Nevertheless, chaonot the relevant thing in the signals of Fig. 8. What ismarkable is the degree of complexity and the robustnesthese waveform structures and the presence of self-similafeatures with respect to the time scale.

Figure 9 shows a numerical simulation obtained frosystem~3! for N56 and other parameters given in the cation. Thecq anddq values have been determined by impoing the occurrence of three Hopf bifurcations on the nopoint atmE552.4, 58.2, and 58.3, with angular frequencequal to 0.02, 125, and 2.98, respectively, and two Hbifurcations on the saddle point atmE564.7 and 104.6, withfrequencies 0.25 and 24.9, respectively. The time evoluof Fig. 9 represents the reflected power formE576 and itsstructure is really similar to that of the 134.8-mW signalFig. 8. The dimensionless characteristic times containedthe numerical evolution are 721, 25, 2.6, 0.26, and 0.05,the corresponding angular frequencies arev j50.009, 0.25,2.4, 24, and 125,j 51, . . . ,5,which must be compared tthe Hopf frequencies. The numerical simulations confithat the attractor evolves around an unstable fixed pointhas effectively performed three successive Hopf bifurcatiat the imposedmE values and with the preselected oscillatifrequencies. The three Hopf frequencies of the node are slar to thev1 , v5 , andv3 of the numerical evolution, respectively. v5 is precisely equal to the Hopf frequency and tslower values ofv1 andv3 can be attributed to the presenof intermediate faster oscillations. On the other hand,attractor visits the neighborhood of a saddle point thatperformed one Hopf bifurcation and is even far from tsecond bifurcation. These Hopf frequencies are very simto the v2 and v4 of the time evolution. Thus the five frequencies contained in the numerical signal are clearly rela

FIG. 9. Five-frequency oscillations numerically obtained from Eqs.~3!for N56; cq54.33,7680,76 400,156 000,11 500,250 anddq54.33,2766,4280,210 400,276,216.7, withq from 1 to 6,c050, andmE576.


sis-ofity

--e

f

n

find

ats

i-

es

r

d

to the Hopf instabilities of the saddle-node pair of fixepoints.

A wider perspective may be achieved by analyzingfull instability behavior of higher-dimensional systems anwith this aim, we present a numerical example forN510 inFig. 10. The dynamical system has been designed by iming five Hopf bifurcations on the node point and four bifucations on the saddle point. FormE5120, the node point isfully unstable, while the saddle point has done only onefurcation and is near to doing the other three bifurcatioThe time evolution shows the nonlinear mixing of nine ocillation modes. The odd label modes are associated with

FIG. 10. Numerical evolution forN510 obtained from Eqs.~3! by impos-ing five Hopf bifurcations on the node point with angular frequencies 0.00.2, 12, 350, and 6000, and four bifurcations on the saddle point withquencies 0.02, 1.8, 70, and 1500. The waveform evolution shows thelinear combination of nine oscillation modes whose frequencies are simto the Hopf values.


oi

hadd

niorithlladdalieso

.rotheolaifurefp

ing

x-:in-rdersl-atedfre-itiesmed

ckeri-caln-bil-is-AL

e-e-esardof

icalesallyvel-ofex-

orthev-thes ofther

aof

ble

ta-ein arityannddethetures isitofof

ix-tio


D

Hopf bifurcations of the node point and the even labels crespond to the saddle point. Self-similarity is clearly seenthe successive zooms of Fig. 10 and it is worth noticing tthe oscillations associated with either the node or the samaintain their roles along the similarity scale.

Experiments and numerical simulations with differemultilayer structures indicate that the full instability behavof the BOITAL systems manifests most generically wtime evolutions like those of Figs. 8–10, where the oscition frequencies associated with either the node or the safixed points clearly play different roles. Nevertheless, qutatively different complex waveforms can also be observfor reduced parameter ranges. For instance, Fig. 11 showexample obtained with a five-layer device similar to thatFig. 8 but with a thinner layer of gel, 160 instead of 180mm.The signal contains five characteristic times of 550, 3.8, 00.25, and 0.1 s, respectively, and the nonlinear mixing pduces a rather irregular waveform where the roles ofdifferent oscillation modes do not appear so clearly definThe numerical studies suggest the association of theserved nongeneric behaviors with the proximity of particueigenvalue degeneracies sustaining codimension-two bcations. For instance, the evolutions of Fig. 11 may belated with a degeneracy of type (0,6 iv) because signals othis type are numerically obtained when one of the Ho

FIG. 11. Experimental example of a time evolution illustrating a sdimensional dynamics with a waveform structure based on five oscillamodes that looks very different to the signals of Figs. 8–10.


r-ntle

t

-le

-danf

7,-ed.b-rr--

f

bifurcations of a fixed point is approached toward the turnpoint of a saddle-node bifurcation.

IV. DISCUSSION AND CONCLUSIONS

First of all let us remark that the BOITAL systems ehibit the full instability behavior with two peculiar features~i! the various oscillation frequencies are rather differentasmuch as they appear roughly scaled for successive oof magnitude, and~ii ! when ordered according to their vaues, the different frequencies appear alternatively associwith either the node or the saddle point and the lowestquency always corresponds to the node. These peculiaroccur because~a! the characteristic times of a given systeare associated with the heat diffusion from the localizsource to the various layers of the cavity spacer, and~b! thedynamical variables participate in the nonlinear feedbathrough a linear combination of them. For instance, numcal simulations with the same model but for nonphysiparameter values yielding more similar oscillation frequecies indicate significant changes in the observed full instaity behaviors, and it is then important to stress that our dcussion here is based on situations like those of the BOITsystems.

The irregular succession of undulations of different frquencies forming the full instability waveforms usually rpeat with regularity and even periodically. This fact indicatthat the high degree of instability represents a way towcreating irregular and complex evolutions independentchaos. In fact, chaos was rarely found during our numersimulations and the higher the number of oscillation modthe more pronounced the absence of chaos. This is resurprising because, according to our interpretation, the deopment of the fully unstable behavior underlies a varietysaddle connections and chaos would then be reasonablypected.

The waveform structures of the full instability behaviare very robust in the sense that they change slowly withcontrol parameter. In other words, the full instability behaior is a coarse phenomenon occurring continuously inparameter space regions where several Hopf bifurcationthe fixed points appear relatively close. This makes anodistinction with respect to chaos, and it explains whymethod exclusively based on the linear stability analysisthe fixed points can be enough for designing fully unstaN-dimensional systems.10

The nonlinear mechanisms responsible for the full insbility waveforms introduce irregularity by affecting both threlative phases and amplitudes of the oscillation modescomplex manner. At the same time, however, the similafeatures typically found in the time evolutions indicateintrinsic organization of the mode mixing mechanism arouthe saddle-node pair of fixed points. The nonlinear momixing may be considered as a global process affectingflow of the phase space region where the complex strucof interrelated invariant sets will be created. The procestriggered by the Hopf bifurcations of the fixed points, butis organized by the global structure of invariant manifoldsthe different saddle limit sets, which underlie a variety

n


odthan

inr-o

oan

thnlu

het t

oveinp

veemri

heera

imi.enth

ndth

safreAupaonnd

arsaounte

enthreno

nthinplre

ccuron-t of

m-eiriple,nson.rre-ofthe

eedon-are

ys--re-

heplea-byri-we

eti-theeri-and

ced

aarc-e-allyos-they,

d inrac-aning

ifm in

ndnt

as98-


D

possible homoclinic and heteroclinic connections. The mmixing happens through a continuous deformation offlow in particular zones of the attractor and, although it mbe accompanied by complex bifurcational sequences, othe bifurcations yielding invariant tori participate directlythe mode mixing. Other bifurcations involving periodic obits, like the period-doubling, cyclic saddle-node, and homclinic bifurcations, do not contain intrinsic mechanisms fthe definition of a new characteristic frequency and they cnot induce mode mixing.

The occurrence of torus bifurcations seems likely infully unstable systems because these systems appear iparameter space relatively close to the loci of eigenvadegeneracies of the type (6 iv1 ,6 iv2 , . . . ,6 ivq), inwhich q Hopf bifurcations happen simultaneously on tsame fixed point, and because it is reasonable to suspeca variety of torus bifurcations can emerge from each onesuch degeneracies,23 The Landau sequence of consecutibifurcations yielding a stable high-dimensional torus isprinciple possible, but we suspect that the correspondingrameter space domain is rather restricted and probablyclose to the corresponding high-order degeneracy. It semore likely the occurrence of different low-dimensional tocreated from the different limit cycles emerged from tfixed point. On the other hand, a given limit cycle can phaps generate successive two-tori with different secondfrequencies, like a fixed point can produce successive lcycles, and the same might happen with the low-order torseems not possible, however, that different limit cycles gerate simultaneously sequences of torus bifurcations withsame set of frequencies. For instance, forN54, the universalunfoldings of (6 iv1 ,6 iv2) show that only one of the twolimit cycles emerging from a node point can do the secoary bifurcation at the frequency of the other cycle but nottwo cycles at once.2

The important point for the full instability behavior ithat the limit cycle that does not do the torus bifurcation calso incorporate an oscillatory component at the otherquency through the mixing mechanism discussed above.the other important point is that not only one fixed point ba set of them, related by saddle-node bifurcations, canticipate together in the generation and mixing of oscillatimodes. These elements constitute the generalized Lascenario where the full instability behavior develops.

The full instability behavior has been investigated in pticular classes of dynamical systems and it is then necesto ask how general the phenomenon is. According tointerpretation, we find reason to suspect that the occurreof the several Hopf bifurcations in a small enough paramedomain will be generically associated with the developmof nonlinear mechanisms of mode mixing. Nevertheless,complexity of the process and the time dynamics featuwill probably change with the actual properties of the nolinear vector field. Particularly relevant may be the valuesthe oscillation frequencies, as well as the order of occurreof the bifurcations on the fixed points. On the other hand,high degree of instability behavior of vector fields possessmultidimensional arrays of fixed points, instead of a simsaddle-node pair, will probably produce rather different


eeyly

-r-

ethee

hatf

a-rys

-ryitIt-e

-e

n-

ndtr-

au

-ryrcertes-f

ceege-

sponses. Such complex structures of fixed points can ofor vector fields based on several linearly independent nlinear functions, such as, for instance, the case of a secoupled nonlinear oscillators.

Another question is which classes of systems are copatible with the full unstable behavior and how can thparameters be adjusted to achieve that behavior. In princthe situations of full instability could be identified by meaof the linear stability analysis of the steady-state solutiNevertheless, in general, it is not easy to establish the cosponding conditions for a given multiparameter familyN-dimensional systems and this probably explains whyfull instability behavior has not already been observed. Whave shown10 that this analysis is feasible for systems bason vector fields whose nonlinear part is a scalar-valued nlinear function of a single variable that, in its turn, is a linecombination of theN dynamic variables, i.e., systems in thform ~3! with an arbitrary nonlinear functionA(c). In thiscase, the linear stability analysis allows us to design the stem in order to obtainN21 Hopf bifurcations on a saddlenode pair of fixed points with preselected values for the fquency and control parameter.

From the phenomenological point of view it is wortnoting that we have found the full instability behavior in thBOITAL devices because they enable us to have a simand effective criterion for properly choosing the set of prameters. In practice, we select the multilayer propertiestrying to see if the alternatively opposite effects of the vaous layers tend to mutually compensate. In other words,attempt to achieve a relative equilibrium among the comptive participation of the various degrees of freedom intononlinear mechanisms. The rule is useful for both the expment and the physical model based on the heat equationallows us to derive proper sets of coefficients for the reduN-order model of Eqs.~1!.

A high degree of Hopf instability behavior requireslarge number of variables participating into the nonlinefeedback by driving competitive effects of different charateristic times. These intrinsic features of the BOITAL dvices may be present in other real-world systems, specithose with a profusion of self-oscillatory processes. Thecillatory behavior is perhaps the most typical response ofevolutionary systems found in biology, economy, ecologand sociology. Such oscillations have probably appearethe course of the system development driven by the intetion with the environment. Extending such a view, we cimagine adaptive systems developing toward exhibithigh-instability states, and this would be in reality the casesuch states would be useful for the existence of the systethe middle of its surroundings.

ACKNOWLEDGMENTS

The authors are indebted to J. I. Rosell, F. Boixader, aR. Pons for their significant contributions in the developmeand understanding of the BOITAL systems. This work wsupported by the Spanish DGES under Grant No. PE0899.


,

s

arf t

e

r-

llyrr

caa

hy

a D

pt.

un.

sec-of

d-

fur-ofciesl.

ns-dlee ei-f theui-the


D

1Dynamic Biological Networks, edited by R. M. Harris-Warrik, E. MarderA. I. Selverton, and M. Moulins~MIT Press, Cambridge, 1992!.

2See, e.g., J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dy-namical Systems, and Bifurcations of Vector Fields~Springer, New York,1983!; S. Wiggins,Introduction to Applied Nonlinear Dynamical Systemand Chaos~Springer, New York, 1990!.

3L. D. Landau, C. R.~Dokl.! Acad. Sci. USSR44, 311~1944!, reproducedin Chaos II, edited by Hao Bai-Lin~World Scientific, Singapore, 1990!, p.115.

4The torus bifurcation does not permit the superposition with arbitrphase assumed by Landau in relation to the occurrence moment osecondary instability.

5See, e.g.,Hydrodynamic Instabilities and the Transition to Turbulenc,edited by H. L. Swinney and J. P. Gollub~Springer-Verlag, Berlin, 1981!.

6D. Ruelle and F. Takens, Commun. Math. Phys.20, 167 ~1971!; S. New-house, D. Ruelle, and F. Takens,ibid. 64, 35 ~1978!.

7Half of the bifurcations occur on each point, if N is odd, while one bifucation more happens on the initially stable point, if N is even.

8A sequence of torus bifurcations initiated from a fixed point is intrinsicarelated to the successive Hopf bifurcations of that point and the cosponding oscillations are usually based on the same frequencies.

9In fact, only one attractor can emerge from the variety of Hopf bifurtions of a saddle-node pair of fixed points but additional attractors marise from secondary processes.

10J. Rius, M. Figueras, R. Herrero, F. Pi, G. Orriols, and J. Farjas, PRev. E62, 333 ~2000!.

11G. Orriols, C. Schmidt, and F. Pi, Opt. Commun.63, 66 ~1987!.12R. Herrero, R. Pons, J. Farjas, F. Pi, and G. Orriols, Phys. Rev. E53, 5627

~1996!.13J. I. Rosell, J. Farjas, R. Herrero, F. Pi, and G. Orriols, Physica D85, 509

~1995!.


yhe

e-

-y

s.

14J. Farjas, J. I. Rosell, R. Herrero, R. Pons, F. Pi, and G. Orriols, Physic95, 107 ~1996!.

15S. L. McCall, Appl. Phys. Lett.32, 284 ~1978!; H. M. Gibbs, S. S. Tang,J. L. Jewell, D. A. Weinberger, and K. Tai,ibid. 41, 221 ~1982!.

16H. J. Kong and W. Y. Hwang, J. Appl. Phys.67, 6066~1990!.17J. I. Rosell, F. Pi, F. Boixader, R. Herrero, F. Farjas, and G. Orriols, O

Commun.82, 162 ~1991!.18R. Herrero, F. Boixader, G. Orriols, J. I. Rosell, and F. Pi, Opt. Comm

113, 324 ~1994!.19P. Glendinning and C. Sparrow, J. Stat. Phys.35, 645 ~1984!.20The two-dimensional eigenspace and the oscillation frequency of the

ondary Hopf bifurcation of the saddle cycle are directly related to thosethe first Hopf bifurcation of the fixed point. This implies that the seconary bifurcation will be subcritical.

21This has been verified by continuously following both periodic orbits.22L. P. Shil’nikov, Int. J. Bifurcation Chaos Appl. Sci. Eng.4, 489 ~1994!.23The possibilities of invariant tori generation through sequences of bi

cations initiated from a fixed point are not well known. The theoryuniversal unfoldings developed for certain low-dimensional degenera~Ref. 2! gives precise information forN up to 4, but only some generaideas may be tentatively extrapolated to higher dimensional problems

24Eighty trajectories initiated from different points of the unstable eigepace of the saddle cycle. The proximity of the initial point to the sadorbit is chosen such that the error due to the separation between thgenspace and the invariant manifold is of the same order as the error onumerical integration. The initial points are taken in 10 temporally eqdistant places along the orbit and with 8 different output angles withintwo-dimensional eigenspace transverse to the orbit.


n-dimensional dynamical systems exploiting instabilities ... · n-dimensional dynamical systems...

Documents