Table Of ContentTitelei Schuster·Just 09.12.2004 16:17 Uhr Seite 3
Heinz Georg Schuster and Wolfram Just
Deterministic Chaos
An Introduction
Fourth, Revised and Enlarged Edition
WILEY-VCH Verlag GmbH & Co. KGaA
Titelei Schuster·Just 09.12.2004 16:17 Uhr Seite 4
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Contents
TableofContents v
Preface ix
ColorPlates xiii
1 Introduction 1
2 ExperimentsandSimpleModels 7
2.1 ExperimentalDetectionofDeterministicChaos . . . . . . . . . . . . . . . 7
2.1.1 DrivenPendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Rayleigh–Be´nardSysteminaBox . . . . . . . . . . . . . . . . . . . 9
2.1.3 StirredChemicalReactions. . . . . . . . . . . . . . . . . . . . . . . 11
2.1.4 He´non–HeilesSystem . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 ThePeriodicallyKickedRotator . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 LogisticMap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.2 He´nonMap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.3 ChirikovMap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 PiecewiseLinearMapsandDeterministicChaos 19
3.1 TheBernoulliShift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 CharacterizationofChaoticMotion . . . . . . . . . . . . . . . . . . . . . . 21
3.2.1 LiapunovExponent . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.2 InvariantMeasure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.3 CorrelationFunction . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 DeterministicDiffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 UniversalBehaviorofQuadraticMaps 33
4.1 ParameterDependenceoftheIterates . . . . . . . . . . . . . . . . . . . . . 35
4.2 PitchforkBifurcationandtheDoublingTransformation . . . . . . . . . . . 37
4.2.1 PitchforkBifurcations . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2.2 Supercycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2.3 DoublingTransformationandα . . . . . . . . . . . . . . . . . . . . 41
4.2.4 LinearizedDoublingTransformationandδ . . . . . . . . . . . . . . 43
vi Contents
4.3 Self-Similarity,UniversalPowerSpectrum,andtheInfluenceofExternalNoise 46
4.3.1 Self-SimilarityinthePositionsoftheCycleElements. . . . . . . . . 46
4.3.2 HausdorffDimension . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3.3 PowerSpectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3.4 InfluenceofExternalNoise . . . . . . . . . . . . . . . . . . . . . . 52
4.4 BehavioroftheLogisticMapforr∞≤r . . . . . . . . . . . . . . . . . . . 55
4.4.1 SensitiveDependenceonParameters. . . . . . . . . . . . . . . . . . 55
4.4.2 StructuralUniversality . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4.3 ChaoticBandsandScaling . . . . . . . . . . . . . . . . . . . . . . . 59
4.5 ParallelsbetweenPeriodDoublingandPhaseTransitions . . . . . . . . . . 61
4.6 ExperimentalSupportfortheBifurcationRoute . . . . . . . . . . . . . . . 64
5 TheIntermittencyRoutetoChaos 69
5.1 MechanismsforIntermittency . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1.1 Type-IIntermittency . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.1.2 LengthoftheLaminarRegion . . . . . . . . . . . . . . . . . . . . . 73
5.2 Renormalization-GroupTreatmentofIntermittency . . . . . . . . . . . . . 75
5.3 Intermittencyand1/f-Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4 ExperimentalObservationoftheIntermittencyRoute . . . . . . . . . . . . 84
5.4.1 DistributionofLaminarLengths . . . . . . . . . . . . . . . . . . . . 84
5.4.2 Type-IIntermittency . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.4.3 Type-IIIIntermittency . . . . . . . . . . . . . . . . . . . . . . . . . 86
6 StrangeAttractorsinDissipativeDynamicalSystems 89
6.1 IntroductionandDefinitionofStrangeAttractors . . . . . . . . . . . . . . . 89
6.1.1 Baker’sTransformation. . . . . . . . . . . . . . . . . . . . . . . . . 92
6.1.2 DissipativeHe´nonMap. . . . . . . . . . . . . . . . . . . . . . . . . 94
6.2 TheKolmogorovEntropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2.1 DefinitionofK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2.2 ConnectionofK totheLiapunovExponents. . . . . . . . . . . . . . 97
6.2.3 AverageTimeoverwhichtheStateofaChaoticSystemcanbePredicted100
6.3 CharacterizationoftheAttractorbyaMeasuredSignal . . . . . . . . . . . 102
6.3.1 ReconstructionoftheAttractorfromaTimeSeries . . . . . . . . . . 103
6.3.2 Generalized Dimensions and Distribution of Singularities in the In-
variantDensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.3.3 GeneralizedEntropiesandFluctuationsaroundtheK-Entropy . . . . 115
6.3.4 Kaplan–YorkeConjecture . . . . . . . . . . . . . . . . . . . . . . . 120
6.4 PicturesofStrangeAttractorsandFractalBoundaries . . . . . . . . . . . . 122
7 TheTransitionfromQuasiperiodicitytoChaos 127
7.1 StrangeAttractorsandtheOnsetofTurbulence . . . . . . . . . . . . . . . . 127
7.1.1 HopfBifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.1.2 Landau’sRoutetoTurbulence . . . . . . . . . . . . . . . . . . . . . 128
7.1.3 Ruelle–Takens–NewhouseRoutetoChaos . . . . . . . . . . . . . . 129
7.1.4 PossibilityofThree-FrequencyQuasiperiodicOrbits . . . . . . . . . 131
7.1.5 Break-upofaTwo-Torus . . . . . . . . . . . . . . . . . . . . . . . . 133
Contents vii
7.2 UniversalPropertiesoftheTransitionfromQuasiperiodicitytoChaos. . . . 136
7.2.1 ModeLockingandtheFareyTree . . . . . . . . . . . . . . . . . . . 139
7.2.2 LocalUniversality . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.2.3 GlobalUniversality . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.3 ExperimentsandCircleMaps . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.3.1 DrivenPendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.3.2 ElectricalConductivityinBariumSodiumNiobate . . . . . . . . . . 153
7.3.3 DynamicsofCardiacCells . . . . . . . . . . . . . . . . . . . . . . . 154
7.3.4 ForcedRayleigh–Be´nardExperiment . . . . . . . . . . . . . . . . . 156
7.4 RoutestoChaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.4.1 Crises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8 RegularandIrregularMotioninConservativeSystems 161
8.1 CoexistenceofRegularandIrregularMotion . . . . . . . . . . . . . . . . . 163
8.1.1 IntegrableSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
8.1.2 PerturbationTheoryandVanishingDenominators . . . . . . . . . . . 165
8.1.3 StableToriandKAMTheorem . . . . . . . . . . . . . . . . . . . . 166
8.1.4 UnstableToriandPoincare´–BirkhoffTheorem . . . . . . . . . . . . 167
8.1.5 HomoclinicPointsandChaos . . . . . . . . . . . . . . . . . . . . . 170
8.1.6 ArnoldDiffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8.1.7 ExamplesofClassicalChaos . . . . . . . . . . . . . . . . . . . . . . 172
8.2 StronglyIrregularMotionandErgodicity . . . . . . . . . . . . . . . . . . . 174
8.2.1 CatMap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8.2.2 HierarchyofClassicalChaos . . . . . . . . . . . . . . . . . . . . . . 176
8.2.3 ThreeClassicalK-Systems . . . . . . . . . . . . . . . . . . . . . . . 180
9 ChaosinQuantumSystems? 183
9.1 TheQuantumCatMap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
9.2 AQuantumParticleinaStadium . . . . . . . . . . . . . . . . . . . . . . . 186
9.3 TheKickedQuantumRotator . . . . . . . . . . . . . . . . . . . . . . . . . 187
10 ControllingChaos 193
10.1 StabilizationofUnstableOrbits . . . . . . . . . . . . . . . . . . . . . . . . 194
10.2 TheOGYMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
10.3 Time-DelayedFeedbackControl . . . . . . . . . . . . . . . . . . . . . . . 199
10.3.1 RhythmicControl. . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
10.3.2 ExtendedTime-DelayedFeedbackControl . . . . . . . . . . . . . . 201
10.3.3 ExperimentalRealizationofTime-DelayedFeedbackControl . . . . 202
10.4 ParametricResonancefromUnstablePeriodicOrbits . . . . . . . . . . . . 203
11 SynchronizationofChaoticSystems 207
11.1 IdenticalSystemswithSymmetricCoupling . . . . . . . . . . . . . . . . . 207
11.1.1 On–OffIntermittency. . . . . . . . . . . . . . . . . . . . . . . . . . 208
11.1.2 Strongvs. WeakSynchronization . . . . . . . . . . . . . . . . . . . 209
11.2 Master–SlaveConfigurations . . . . . . . . . . . . . . . . . . . . . . . . . 210
viii Contents
11.3 GeneralizedSynchronization . . . . . . . . . . . . . . . . . . . . . . . . . 212
11.3.1 StrangeNonchaoticAttractors . . . . . . . . . . . . . . . . . . . . . 212
11.4 PhaseSynchronizationofChaoticSystems . . . . . . . . . . . . . . . . . . 213
12 SpatiotemporalChaos 217
12.1 ModelsforSpace–TimeChaos . . . . . . . . . . . . . . . . . . . . . . . . 217
12.1.1 CoupledMapLattices . . . . . . . . . . . . . . . . . . . . . . . . . 217
12.1.2 CoupledOscillatorModels . . . . . . . . . . . . . . . . . . . . . . . 218
12.1.3 ComplexGinzburg–LandauEquation . . . . . . . . . . . . . . . . . 220
12.1.4 Kuramoto–SivashinskyEquation . . . . . . . . . . . . . . . . . . . . 220
12.2 CharacterizationofSpace–TimeChaos . . . . . . . . . . . . . . . . . . . . 221
12.2.1 LiapunovSpectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
12.2.2 Co-movingLiapunovExponent . . . . . . . . . . . . . . . . . . . . 223
12.2.3 ChronotopicLiapunovAnalysis . . . . . . . . . . . . . . . . . . . . 224
12.3 NonlinearNonequilibriumSpace–TimeDynamics . . . . . . . . . . . . . . 225
12.3.1 FullyDevelopedTurbulence . . . . . . . . . . . . . . . . . . . . . . 225
12.3.2 SpatiotemporalIntermittency. . . . . . . . . . . . . . . . . . . . . . 227
12.3.3 MolecularDynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Outlook 231
Appendix 233
A DerivationoftheLorenzModel . . . . . . . . . . . . . . . . . . . . . . . . 233
B StabilityAnalysisandtheOnsetofConvectionandTurbulenceintheLorenz
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
C TheSchwarzianDerivative . . . . . . . . . . . . . . . . . . . . . . . . . . 236
D RenormalizationoftheOne-DimensionalIsingModel . . . . . . . . . . . . 238
E DecimationandPathIntegralsforExternalNoise . . . . . . . . . . . . . . 240
F Shannon’sMeasureofInformation . . . . . . . . . . . . . . . . . . . . . . 243
F.1 InformationCapacityofaStore . . . . . . . . . . . . . . . . . . . . 243
F.2 InformationGain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
G PeriodDoublingfortheConservativeHe´nonMap . . . . . . . . . . . . . . 245
H UnstablePeriodicOrbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
RemarksandReferences 257
Index 283
Preface
Since1994whenthelasteditionofthepresentmonographwaspublished,thefieldofNonlin-
earSciencehasdevelopedtremendously. Itisnowadaysnolongerpossibletogiveacompre-
hensibleintroductioninto,andabalancedoverviewofthedifferentbrancheswithinthisfield.
Followingthegeneralpracticeofthepreviouseditionsitisthescopeofthisfourthaugmented
edition to introduce aspects of Nonlinear Dynamics at a level which is accessible to a wide
audience. Wehaveintensifiedandaddedthreenewtopics:
– ControlofchaosisoneofthemostpopularbranchesofNonlinearScience. Asapartic-
ularnewaspectwehaveincludedacomprehensivediscussionoftime-delayedfeedback
controlwhichiswidelyusedinapplications.
– Topicsinsynchronizationbecamerecentlyquitepopularfromafundamentalaswellas
anappliedpointofview. Weintroducebasicconceptsaswellasnovelnotionslikephase
synchronizationorstrangenonchaotic,attractors,atanelementarylevel.
– Spatiotemporalchaoscoversawiderangeoftopics,fromclassicalfieldsinphysicssuch
as hydrodynamics to current research topics in theoretical biophysics, which are com-
monlyrelatedwiththenonlineardynamicsofalargenumberofdegreesoffreedom. We
introduceherebasicfeaturesofrelevantmodelsystemsaswellasselectedconceptsfor
quantitativeanalysis. Butourexpositionisfarfromcomplete.
The fourth edition benefits from data and figures that have been provided by several col-
leagues, inparticularbyR.Klages, J.Kurths, A.Pikovski, H.Posch, andM.Rosenblum. It
is a pleasure to thank E. Scho¨ll for his kind hospitality during a stay at Berlin University of
Technology,wherepartsoftheneweditionwerewritten. Weareindebtedtothepublisher,in
particulartoDr. M.Ba¨randR.Schulz, fortheircontinualhelpinpreparingthemanuscript.
Despite the remarkable support from various people the present edition could still contain
mistakes. Weapologizeinadvanceforsuchinconsistenciesandweinvitethereadertoreport
tousanydeficiencies.
Kiel/London,October2004
H.G.Schuster,W.Just
x Preface
PrefacetotheThirdEdition
Since the last edition of this book in 1989 the field of deterministic chaos has continued to
grow. Withinthewealthofnewresultstherearethreemajortrends.
– Unstableperiodicorbitshavebeenrediscoveredasbuildingblocksofchaoticdynamics,
especiallythroughtheworkofCvitanovichetal. (1990). Theydevelopedanexpansion
ofphysicalaveragesintermsofprimitivecycles(seealsoAppendixH).
– Exploitingtheconceptofunstableperiodicorbits,Ott,GrebogiandYorkedemonstrated
in1990thatdeterministicchaoscanbecontrolled.Theyfoundthatsmalltime-dependent
changesinthecontrolparameterofthesystemcanstabilizepreviouslyunstableperiodic
cyclesinsuchawaythatthesystembecomesnonchaotic(seeChapter10).
– Therearenewtheoreticalandexperimentalresultsinthefieldofquantumchaoswhich
aredescribedexcellentlyinthenewbooksbyGutzwiller(1990),Haake(1991)andRe-
ichl(1992).
During the preparation of the new edition, J. C. Gruel helped with the pictures of the new
chapter, Mrs. H. Heimann typed the new text, M. Poulson and R. Wengenmayr from VCH
Publishers took care of the editorial work. H. J. Stockmann and H. J. Stein contributed the
fascinatingpicturesofsimulationsofquantumchaosinmicrowaveresonators. Iwouldliketo
thankallthesepeoplefortheircooperationandpatience.
Kiel,August1994
H.G.Schuster
PrefacetotheSecondEdition
Thisisarevisedandupdatedversionofthefirstedition, towhichnewsectionsonsensitive
parameterdependence,fatfractals,characterizationofattractorsbyscalingindices,theFarey
tree, and the notion of global universality have been added. I thank P. C. T. de Boer, J. L.
Grant, P. Grassberger, W. Greulich, F. Kaspar, K. Pawelzik, K. Schmidt, and S. Smid for
helpfulhintsandremarks,andMrs. AdlfingerandMrs. Boffofortheirpatienthelpwiththe
manuscript.
Kiel,August1987
H.G.Schuster
PrefacetotheFirstEdition
Daily experience shows that, for many physical systems, small changes in the initial condi-
tionsleadtosmallchangesintheoutcome. Ifwedriveacarandturnthesteeringwheelonly
alittle,ourcoursewilldifferonlyslightlyfromthatwhichthecarwouldhavetakenwithout
thischange. Buttherearecasesforwhichtheoppositeofthisruleistrue: Foracoinwhichis
placedonitsrim,aslighttouchissufficienttodeterminethesideonwhichitwillfall. Thus
Preface xi
thesequenceofheadsandtailswhichweobtainwhentossingacoinexhibitsanirregularor
chaotic behavior in time, because extremely small changes in the initial conditions can lead
tocompletelydifferentoutcomes. Ithasbecomeclearinrecentyears,partlytriggeredbythe
studiesofnonlinearsystemsusinghigh-speedcomputers,thatasensitivedependenceonthe
initialconditions,whichresultsinachaotictimebehavior,isbynomeansexceptionalbutis
a typical property of many systems. Such behavior has, for example, been found in period-
ically stimulated cardiac cells, in electronic circuits, at the onset of turbulence in fluids and
gases,inchemicalreactions,inlasers,etc. Mathematically,allnonlineardynamicalsystems
withmorethantwodegreesoffreedom, i.e.,, especiallymanybiological, meteorologicalor
economic models, can display chaos and, therefore, become unpredictable over longer time
scales.“Deterministicchaos”isnowaveryactivefieldofresearchwithmanyexcitingresults.
Methodshavebeendevelopedtoclassifydifferenttypesofchaos,andithasbeendiscovered
that many systems show, as a function of an external control parameter, similar transitions
from order to chaos. This universal behavior is reminiscent of ordinary second-order phase
transitions, andtheintroductionofrenormalizationandscalingmethodsfromstatisticalme-
chanics has brought new perspectives into the study of deterministic chaos. It is the aim of
thisbooktoprovideaself-containedintroductiontothisfieldfromaphysicist’spointofview.
Thebookgrewoutofaseriesoflectures,whichIgaveduringthesummertermsof1982and
1983 at the University of Frankfurt, and it requires no knowledge which a graduate student
in physics would not have. A glance at the table of contents shows that new concepts such
astheKolmogoroventropy,strangeattractors,etc.,ornewtechniquessuchasthefunctional
renormalizationgroup,areintroducedatanelementarylevel. Ontheotherhand,Ihopethat
there is enough material for research workers who want to know, for example, how deter-
ministic chaos can be distinguished experimentally from white noise, or who want to learn
howtoapplytheirknowledgeaboutequilibriumphasetransitionstothestudyof(nonequilib-
rium) transitions from order to chaos. During the preparation of this book the manuscripts,
preprintsanddiscussion,theremarksofG.Eilenberger,K.Kehr,H.Leschke,W.Selke,and
M. Schmutz were of great help. P. Berge, M. Dubois, W. Lauterborn, W. Martienssen, G.
Pfisterandtheircoworkerssuppliedseveral,partlyunpublished,picturesoftheirexperiments.
H. O. Peitgen, P. H. Richter and their group gave permission to include some of their most
fascinatingcomputerpicturesinthisbook(seecoverandSection6.4). Allcontributionsare
gratefullyappreciated.Furthermore,IwanttothankW.Greulich,D.Hackenbracht,M.Heise,
L.L.Hirst,R.Liebmann,I.Neil,andespeciallyI.Procacciaforcarefullyreadingpartsofthe
manuscript and for useful criticism and comments. I also acknowledge illuminating discus-
sionswithV.Emery,P.Grassberger,D.Grempel,S.Grossmann,S.Fishman,andH.Horner.
It is a pleasure to thank R. Hornreich for the kind hospitality extended to me during a stay
attheWeizmannInstitute,whereseveralchaptersofthisbookwerewritten,withthesupport
oftheMinervafoundation. Lastbutnotleast, IthankMrs. BoffoandMrs. Knollefortheir
excellentassistanceinpreparingtheillustrationsandthetext.
Frankfurt,October1984
H.G.Schuster
Legends to Plates I–XX
ManyoftheseplatesarepartofChapter6. Accordingly,referencesmentionedinthelegends
aretobefoundonpages89–125.
I. Biperiodic flow in a Be´nard experiment. Figs. 1–8 show interferometric pictures of a
Be´nardcellinthebiperiodicregime; thatis, therearetwoincommensuratefrequencies
inthepowerspectrum(seealsopages2.1.2–2.1.2).Thetimebetweensuccessivepictures
is10s. Thefirstperiodlasts40safterwhichthe“mouth”inthemiddleofthepictures
repeatsitself(seeFigs.1and5).Butthedetails,e.g.,intheupperrightcornersofFigs.1
and5arenotthesame;thatis,themotionisnotsimplyperiodic. (Fromafilmtakenby
P.BergeandM.Dubois,CENSaclay,Gif-sur-Yvette,France.) [pagexv]
II. Nonlinearelectronicoscillator (seealsoFig.46onpage66):Thecurrent-versus-voltage
phaseportraits(atthenonlineardiode)areshownontheoscilloscopescreen.Forincreas-
ingdrivingvoltageoneobservestheperiod-doublingroute.Thenonlinearityofthediode
thathasbeenusedinthisexperimentdiffersfromeq.4.119. (PicturetakenbyW.Meyer-
Ilse,afterKlinkeretal.,1984.) [pagexvi]
III. Taylor instability. a) Formation of rolls, b) the rolls start oscillating, c) a more com-
plicated oscillatory motion, d) chaos. (After Pfister, 1984; see also pages 133–136.)
[pagexvi]
IV. Disturbed heartbeats. The voltage difference (black) across the cell membrane of one
cellofanaggregateofheartcellsfromembryonicchickenshowsa)phaselockingwith
thestimulatingpulseandb)irregulardynamics,displayingescapeorinterpolationbeats
if the time between successive periodic stimuli (red) is changed from 240 ms in a) to
560msinb). (AfterGlassetal.,1983;seealsopage7.3.3.) [pagexvii]
V. ChaoticelectricalconductioninBSNcrystals. Thebirefringencepatternofaferroelec-
tricBSNcrystalshowsdomainwallswhichmirrorthechargetransportneartheonsetof
chaos(seealsoFig.116onpage153). Forclarity, thedarklinesintheoriginalpattern
havebeenredrawninred. (AfterMartinetal.,1984.) [pagexvii]
VI. Power spectra of cavitation noise: The noise amplitude is depicted in colors, and the
inputpressureisincreasinglinearlywithtime. Oneobserves(withincreasingpressure)
asubharmonicroute f → f /2→ f /4→chaos. Thispictureisthecoloredversionof
0 0 0
Fig. 47c on page 67. (Picture taken by E. Suchla, after Lauterborn and Cramer, 1981.)
[pagexviii]
VII. TheCassinidivision. TheringofSaturn(a)showsamajorgap(b),theso-calledCassini
division, because the motion on this orbit is unstable (see also Fig. 142 on page 174).
DeterministicChaos:AnIntroduction,FourthEdition.H.-G.SchusterandW.Just
Copyright(cid:1)c2005WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
ISBN:3-527-40415-5
