The Topology of Chaos: Alice in Stretch and Squeezeland, Second Edition

609 Pages·2011·22.56 MB·English

by  Prof. Robert Gilmore

RobertGilmoreandMarcLefranc
TheTopologyofChaos
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Robert Gilmore and Marc Lefranc
The Topology of Chaos
Alice in Stretch and Squeezeland
Second Revisedand EnlargedEdition
WILEY-VCH Verlag GmbH & Co. KGaA
TheAuthors AllbookspublishedbyWiley-VCHarecarefully
produced.Nevertheless,authors,editors,and
Prof.RobertGilmore publisherdonotwarranttheinformation
DrexelUniversity containedinthesebooks,includingthisbook,to
Dept.ofPhysics befreeoferrors.Readersareadvisedtokeepin
Philadelphia,USA mindthatstatements,data,illustrations,
[email protected] proceduraldetailsorotheritemsmay
inadvertentlybeinaccurate.
Prof.MarcLefranc
LaboratoiredePhysiquedesLasers,Atomes, LibraryofCongressCardNo.:appliedfor
Molécules
UniversitédesSciencesetTechnologiesdeLille BritishLibraryCataloguing-in-PublicationData:
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[email protected]
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V
Contents
PrefacetoSecondEdition XVII
PrefacetotheFirstEdition XIX
1 Introduction 1
1.1 BriefReviewofUsefulConcepts 2
1.2 LaserwithModulatedLosses 4
1.3 ObjectivesofaNewAnalysisProcedure 11
1.4 PreviewofResults 12
1.5 OrganizationofThisWork 14
2 DiscreteDynamicalSystems:Maps 19
2.1 Introduction 19
2.2 LogisticMap 20
2.3 BifurcationDiagrams 22
2.4 ElementaryBifurcationsintheLogisticMap 25
2.4.1 Saddle–NodeBifurcation 25
2.4.2 Period-DoublingBifurcation 29
2.5 MapConjugacy 32
2.5.1 ChangesofCoordinates 32
2.5.2 InvariantsofConjugacy 33
2.6 FullyDevelopedChaosintheLogisticMap 34
2.6.1 IteratesoftheTentMap 35
2.6.2 LyapunovExponents 36
2.6.3 SensitivitytoInitialConditionsandMixing 37
2.6.4 ChaosandDensityof(Unstable)PeriodicOrbits 38
2.6.4.1 NumberofPeriodicOrbitsoftheTentMap 38
2.6.4.2 ExpansivenessImpliesInfinitelyManyPeriodicOrbits 39
2.6.5 SymbolicCodingofTrajectories:FirstApproach 40
2.7 One-DimensionalSymbolicDynamics 42
2.7.1 Partitions 42
2.7.2 SymbolicDynamicsofExpansiveMaps 44
2.7.3 GrammarofChaos:FirstApproach 48
2.7.3.1 IntervalArithmeticsandInvariantInterval 48
VI Contents
2.7.3.2 ExistenceofForbiddenSequences 49
2.7.4 KneadingTheory 51
2.7.4.1 OrderingofItineraries 52
2.7.4.2 AdmissibleSequences 54
2.7.5 BifurcationDiagramoftheLogisticMapRevisited 55
2.7.5.1 Saddle–NodeBifurcations 55
2.7.5.2 Period-DoublingBifurcations 56
2.7.5.3 UniversalSequence 57
2.7.5.4 Self-SimilarStructureoftheBifurcationDiagram 58
2.8 ShiftDynamicalSystems,MarkovPartitions,andEntropy 59
2.8.1 ShiftsofFiniteTypeandTopologicalMarkovChains 59
2.8.2 PeriodicOrbitsandTopologicalEntropyofaMarkovChain 61
2.8.3 MarkovPartitions 63
2.8.4 ApproximationbyMarkovChains 65
2.8.5 ZetaFunction 65
2.8.6 DealingwithGrammars 66
2.8.6.1 SimpleGrammars 67
2.8.6.2 ComplicatedGrammars 69
2.9 FingerprintsofPeriodicOrbitsandOrbitForcing 70
2.9.1 PermutationofPeriodicPointsasaTopologicalInvariant 70
2.9.2 TopologicalEntropyofaPeriodicOrbit 72
2.9.3 Period3ImpliesChaosandSarkovskii’sTheorem 74
2.9.4 Period3DoesNotAlwaysImplyChaos:
RoleofPhase-SpaceTopology 75
2.9.5 PermutationsandOrbitForcing 75
2.10 Two-DimensionalDynamics:Smale’sHorseshoe 77
2.10.1 HorseshoeMap 77
2.10.2 SymbolicDynamicsoftheInvariantSet 78
2.10.3 DynamicalProperties 81
2.10.4 VariationsontheHorseshoeMap:BakerMaps 82
2.11 HénonMap 85
2.11.1 AOnce-FoldingMap 85
2.11.2 SymbolicDynamicsoftheHénonMap:Coding 87
2.11.3 SymbolicDynamicsoftheHénonMap:Grammar 93
2.12 CircleMaps 96
2.12.1 ANewGlobalTopology 96
2.12.2 FrequencyLockingandArnoldTongues 96
2.12.3 ChaoticCircleMapsasLimitsofAnnulusMaps 100
2.13 AnnulusMaps 100
2.14 Summary 104
3 ContinuousDynamicalSystems:Flows 105
3.1 DefinitionofDynamicalSystems 105
3.2 ExistenceandUniquenessTheorem 106
3.3 ExamplesofDynamicalSystems 107
Contents VII
3.3.1 DuffingEquation 107
3.3.2 VanderPolEquation 109
3.3.3 LorenzEquations 111
3.3.4 RösslerEquations 113
3.3.5 ExamplesofNondynamicalSystems 114
3.3.5.1 EquationwithNon-LipschitzForcingTerms 115
3.3.5.2 DelayDifferentialEquations 115
3.3.5.3 StochasticDifferentialEquations 116
3.3.6 AdditionalObservations 117
3.4 ChangeofVariables 120
3.4.1 Diffeomorphisms 120
3.4.2 Examples 121
3.4.3 StructureTheory 124
3.5 FixedPoints 125
3.5.1 DependenceonTopologyofPhaseSpace 125
3.5.2 HowtoFindFixedPointsinRn 126
3.5.3 BifurcationsofFixedPoints 127
3.5.4 StabilityofFixedPoints 130
3.6 PeriodicOrbits 131
3.6.1 LocatingPeriodicOrbitsinRn(cid:2)1(cid:2)S1 131
3.6.2 BifurcationsofFixedPoints 132
3.6.3 StabilityofFixedPoints 133
3.7 FlowsNearNonsingularPoints 134
3.8 VolumeExpansionandContraction 136
3.9 StretchingandSqueezing 137
3.10 TheFundamentalIdea 138
3.11 Summary 139
4 TopologicalInvariants 141
4.1 StretchingandSqueezingMechanisms 141
4.2 LinkingNumbers 145
4.2.1 Definitions 146
4.2.2 ReidemeisterMoves 147
4.2.3 Braids 148
4.2.4 Examples 151
4.2.5 LinkingNumbersforaHorseshoe 153
4.2.6 LinkingNumbersfortheLorenzAttractor 154
4.2.7 LinkingNumbersforthePeriod-DoublingCascade 154
4.2.8 LocalTorsion 155
4.2.9 WritheandTwist 156
4.2.10 AdditionalProperties 158
4.3 RelativeRotationRates 159
4.3.1 Definition 160
4.3.2 ComputingRelativeRotationRates 160
4.3.3 HorseshoeMechanism 163
VIII Contents
4.3.4 AdditionalProperties 168
4.4 RelationbetweenLinkingNumbersandRelativeRotationRates 169
4.5 AdditionalUsesofTopologicalInvariants 170
4.5.1 BifurcationOrganization 170
4.5.2 TorusOrbits 171
4.5.3 AdditionalRemarks 171
4.6 Summary 174
5 BranchedManifolds 175
5.1 ClosedLoops 175
5.2 WhatDoesThisHavetoDowithDynamicalSystems? 178
5.3 GeneralPropertiesofBranchedManifolds 178
5.4 Birman–WilliamsTheorem 181
5.4.1 Birman–WilliamsProjection 182
5.4.2 StatementoftheTheorem 183
5.5 RelaxationofRestrictions 184
5.5.1 StronglyContractingRestriction 184
5.5.2 HyperbolicRestriction 185
5.6 ExamplesofBranchedManifolds 186
5.6.1 Smale–RösslerSystem 186
5.6.2 LorenzSystem 188
5.6.3 DuffingSystem 189
5.6.4 VanderPolSystem 192
5.7 UniquenessandNonuniqueness 194
5.7.1 LocalMoves 195
5.7.2 GlobalMoves 197
5.8 StandardForm 200
5.9 TopologicalInvariants 201
5.9.1 KneadingTheory 202
5.9.2 LinkingNumbers 205
5.9.3 RelativeRotationRates 207
5.10 AdditionalProperties 207
5.10.1 PeriodasLinkingNumber 208
5.10.2 EBK-LikeExpressionforPeriods 208
5.10.3 PoincaréSection 209
5.10.4 Blow-UpofBranchedManifolds 210
5.10.5 Branched-ManifoldSingularities 211
5.10.6 ConstructingaBranchedManifoldfromaMap 212
5.10.7 TopologicalEntropy 213
5.11 Subtemplates 216
5.11.1 TwoAlternatives 216
5.11.2 AChoice 218
5.11.3 TopologicalEntropy 219
5.11.4 SubtemplatesoftheSmaleHorseshoe 221
Contents IX
5.11.5 SubtemplatesInvolvingTongues 222
5.12 Summary 224
6 TopologicalAnalysisProgram 227
6.1 BriefSummaryoftheTopologicalAnalysisProgram 227
6.2 OverviewoftheTopologicalAnalysisProgram 228
6.2.1 FindPeriodicOrbits 228
6.2.2 EmbedinR3 229
6.2.3 ComputeTopologicalInvariants 230
6.2.4 IdentifyTemplate 230
6.2.5 VerifyTemplate 231
6.2.6 ModelDynamics 232
6.2.7 ValidateModel 233
6.3 Data 234
6.3.1 DataRequirements 235
6.3.2 ProcessingintheTimeDomain 236
6.3.3 ProcessingintheFrequencyDomain 238
6.3.3.1 High-FrequencyFilter 238
6.3.3.2 Low-FrequencyFilter 238
6.3.3.3 DerivativesandIntegrals 239
6.3.3.4 HilbertTransforms 240
6.3.3.5 FourierInterpolation 241
6.3.3.6 TransformandInterpolation 242
6.4 Embeddings 243
6.4.1 EmbeddingsforPeriodicallyDrivenSystems 244
6.4.2 DifferentialEmbeddings 244
6.4.3 Differential–IntegralEmbeddings 247
6.4.4 EmbeddingswithSymmetry 248
6.4.5 Time-DelayEmbeddings 249
6.4.6 Coupled-OscillatorEmbeddings 251
6.4.7 SVDProjections 252
6.4.8 SVDEmbeddings 254
6.4.9 EmbeddingTheorems 254
6.5 PeriodicOrbits 256
6.5.1 CloseReturnsPlotsforFlows 256
6.5.1.1 CloseReturnsHistograms 258
6.5.1.2 TestsforChaos 258
6.5.2 CloseReturnsinMaps 259
6.5.2.1 FirstReturnMap 259
6.5.2.2 pthReturnMap 260
6.5.3 MetricMethods 261
6.6 ComputationofTopologicalInvariants 262
6.6.1 EmbedOrbits 262
6.6.2 LinkingNumbersandRelativeRotationRates 262
6.6.3 LabelOrbits 263
X Contents
6.7 IdentifyTemplate 263
6.7.1 Period-1andPeriod-2Orbits 263
6.7.2 MissingOrbits 264
6.7.3 MoreComplicatedBranchedManifolds 264
6.8 ValidateTemplate 264
6.8.1 PredictAdditionalToplogicalInvariants 265
6.8.2 Compare 265
6.8.3 GlobalProblem 265
6.9 ModelDynamics 265
6.10 ValidateModel 268
6.10.1 QualitativeValidation 269
6.10.2 QuantitativeValidation 269
6.11 Summary 270
7 FoldingMechanisms:A 271
2
7.1 Belousov–ZhabotinskiiChemicalReaction 272
7.1.1 LocationofPeriodicOrbits 273
7.1.2 EmbeddingAttempts 274
7.1.3 TopologicalInvariants 278
7.1.4 Template 281
7.1.5 DynamicalProperties 281
7.1.6 Models 283
7.1.7 ModelVerification 283
7.2 LaserwithSaturableAbsorber 285
7.3 StringedInstrument 288
7.3.1 ExperimentalArrangement 288
7.3.2 FlowModels 290
7.3.3 DynamicalTests 291
7.3.4 TopologicalAnalysis 291
7.4 LaserswithLow-IntensitySignals 294
7.4.1 SVDEmbedding 295
7.4.2 TemplateIdentification 296
7.4.3 ResultsoftheAnalysis 297
7.5 TheLasersinLille 297
7.5.1 ClassBLaserModel 298
7.5.2 CO LaserwithModulatedLosses 304
2
7.5.3 Nd-DopedYAGLaser 308
7.5.4 Nd-DopedFiberLaser 311
7.5.5 SynthesisofResults 318
7.6 TheLaserinZaragoza 322
7.7 NeuronwithSubthresholdOscillations 328
7.8 Summary 334
8 TearingMechanisms:A 337
3
8.1 LorenzEquations 337
8.1.1 FixedPoints 338