|
|
| Preface
| v
|
|
| 1
| Introduction
| 1
|
| 1.1
| Laser with Modulated Losses
| 2
|
| 1.2
| Objectives of a New Analysis Procedure
| 10
|
| 1.3
| Preview of Results
| 11
|
| 1.4
| Organization of This Work
| 12
|
|
| 2
| Discrete Dynamical Systems: Maps
| 17
|
| 2.1
| Introduction
| 17
|
| 2.2
| Logistic Map
| 19
|
| 2.3
| Bifurcation Diagrams
| 21
|
| 2.4
| Elementary Bifurcations in the Logistic Map
| 23
|
| 2.4.1
| Saddle-Node Bifurcation
| 23
|
| 2.4.2
| Period-Doubling Bifurcation
| 27
|
| 2.5
| Map Conjugacy
| 30
|
| 2.5.1
| Changes of Coordinates
| 30
|
| 2.5.2
| Invariants of Conjugacy
| 31
|
| 2.6
| Fully Developed Chaos in the Logistic Map
| 32
|
| 2.6.1
| Iterates of the Tent Map
| 33
|
| 2.6.2
| Lyapunov Exponents
| 35
|
| 2.6.3
| Sensitivity to Initial Conditions and Mixing
| 35
|
| 2.6.4
| Chaos and Density of (Unstable) Periodic Orbits
| 36
|
| 2.6.5
| Symbolic Coding of Trajectories: First Approach
| 38
|
| 2.7
| One-Dimensional Symbolic Dynamics
| 40
|
| 2.7.1
| Partitions
| 40
|
| 2.7.2
| Symbolic Dynamics of Expansive Maps
| 43
|
| 2.7.3
| Grammar of Chaos: First Approach
| 46
|
| 2.7.4
| Kneading Theory
| 49
|
| 2.7.5
| Bifurcation Diagram of the Logistic Map Revisited
| 53
|
| 2.8
| Shift Dynamical Systems, Markov Partitions, and Entropy
| 57
|
| 2.8.1
| Shifts of Finite Type and Topological Markov Chains
| 57
|
| 2.8.2
| Periodic Orbits and Topological Entropy of a Markov Chain
| 59
|
| 2.8.3
| Markov Partitions
| 61
|
| 2.8.4
| Approximation by Markov Chains
| 62
|
| 2.8.5
| Zeta Function
| 63
|
| 2.8.6
| Dealing with Grammars
| 64
|
| 2.9
| Fingerprints of Periodic Orbits and Orbit Forcing
| 67
|
| 2.9.1
| Permutation of Periodic Points as a Topological Invariant
| 67
|
| 2.9.2
| Topological Entropy of a Periodic Orbit
| 69
|
| 2.9.3
| Period-3 Implies Chaos and Sarkovskii's Theorem
| 71
|
| 2.9.4
| Period-3 Does Not Always Imply Chaos: Role of Phase-Space Topology
| 72
|
| 2.9.5
| Permutations and Orbit Forcing
| 72
|
| 2.10
| Two-Dimensional Dynamics: Smale's Horseshoe
| 74
|
| 2.10.1
| Horseshoe Map
| 74
|
| 2.10.2
| Symbolic Dynamics of the Invariant Set
| 75
|
| 2.10.3
| Dynamical Properties
| 78
|
| 2.10.4
| Variations on the Horseshoe Map: Baker Maps
| 79
|
| 2.11
| Hénon Map
| 82
|
| 2.11.1
| A Once-Folding Map
| 82
|
| 2.11.2
| Symbolic Dynamics of the Hénon Map
| 84
|
| 2.12
| Circle Maps
| 90
|
| 2.12.1
| A New Global Topology
| 90
|
| 2.12.2
| Frequency Locking and Arnol'd Tongues
| 91
|
| 2.12.3
| Chaotic Circle Maps and Annulus Maps
| 94
|
| 2.13
| Summary
| 95
|
|
| 3
| Continuous Dynamical Systems: Flows
| 97
|
| 3.1
| Definition of Dynamical Systems
| 97
|
| 3.2
| Existence and Uniqueness Theorem
| 98
|
| 3.3
| Examples of Dynamical Systems
| 99
|
| 3.3.1
| Duffing Equation
| 99
|
| 3.3.2
| van der Pol Equation
| 100
|
| 3.3.3
| Lorenz Equations
| 102
|
| 3.3.4
| Rossler Equations
| 105
|
| 3.3.5
| Examples of Nondynamical Systems
| 106
|
| 3.3.6
| Additional Observations
| 109
|
| 3.4
| Change of Variables
| 112
|
| 3.4.1
| Diffeomorphisms
| 112
|
| 3.4.2
| Examples
| 112
|
| 3.4.3
| Structure Theory
| 114
|
| 3.5
| Fixed Points
| 116
|
| 3.5.1
| Dependence on Topology of Phase Space
| 116
|
| 3.5.2
| How to Find Fixed Points in Rn
| 117
|
| 3.5.3
| Bifurcations of Fixed Points
| 118
|
| 3.5.4
| Stability of Fixed Points
| 120
|
| 3.6
| Periodic Orbits
| 121
|
| 3.6.2
| Bifurcations of Fixed Points
| 122
|
| 3.6.3
| Stability of Fixed Points
| 123
|
| 3.7
| Flows near Nonsingular Points
| 124
|
| 3.8
| Volume Expansion and Contraction
| 125
|
| 3.9
| Stretching and Squeezing
| 126
|
| 3.10
| The Fundamental Idea
| 127
|
| 3.11
| Summary
| 128
|
|
| 4
| Topological Invariants
| 131
|
| 4.1
| Stretching and Squeezing Mechanisms
| 132
|
| 4.2
| Linking Numbers
| 136
|
| 4.2.1
| Definitions
| 136
|
| 4.2.2
| Reidemeister Moves
| 138
|
| 4.2.3
| Braids
| 139
|
| 4.2.4
| Examples
| 142
|
| 4.2.5
| Linking Numbers for the Horseshoe
| 143
|
| 4.2.6
| Linking Numbers for the Lorenz Attractor
| 144
|
| 4.2.7
| Linking Numbers for the Period-Doubling Cascade
| 146
|
| 4.2.8
| Local Torsion
| 146
|
| 4.2.9
| Writhe and Twist
| 147
|
| 4.2.10
| Additional Properties
| 148
|
| 4.3
| Relative Rotation Rates
| 149
|
| 4.3.1
| Definition
| 150
|
| 4.3.2
| How to Compute Relative Rotation Rates
| 151
|
| 4.3.3
| Horseshoe Mechanism
| 155
|
| 4.3.4
| Additional Properties
| 159
|
| 4.4
| Relation between Linking Numbers and Relative Rotation Rates
| 159
|
| 4.5
| Additional Uses of Topological Invariants
| 160
|
| 4.5.1
| Bifurcation Organization
| 160
|
| 4.5.2
| Torus Orbits
| 161
|
| 4.5.3
| Additional Remarks
| 161
|
| 4.6
| Summary
| 164
|
|
| 5
| Branched Manifolds
| 165
|
| 5.1
| Closed Loops
| 166
|
| 5.1.1
| Undergraduate Students
| 166
|
| 5.1.2
| Graduate Students
| 166
|
| 5.1.3
| The Ph.D. Candidate
| 166
|
| 5.1.4
| Important Observation
| 168
|
| 5.2
| What Has This Got to Do with Dynamical Systems?
| 169
|
| 5.3
| General Properties of Branched Manifolds
| 169
|
| 5.4
| Birman--Williams Theorem
| 171
|
| 5.4.1
| Birman--Williams Projection
| 171
|
| 5.4.2
| Statement of the Theorem
| 173
|
| 5.5
| Relaxation of Restrictions
| 175
|
| 5.5.1
| Strongly Contracting Restriction
| 175
|
| 5.5.2
| Hyperbolic Restriction
| 176
|
| 5.6
| Examples of Branched Manifolds
| 176
|
| 5.6.1
| Smale--Rossler System
| 177
|
| 5.6.2
| Lorenz System
| 179
|
| 5.6.3
| Duffing System
| 180
|
| 5.6.4
| van der Pol System
| 182
|
| 5.7
| Uniqueness and Nonuniqueness
| 186
|
| 5.7.1
| Local Moves
| 186
|
| 5.7.2
| Global Moves
| 187
|
| 5.8
| Standard Form
| 190
|
| 5.9
| Topological Invariants
| 193
|
| 5.9.1
| Kneading Theory
| 193
|
| 5.9.2
| Linking Numbers
| 197
|
| 5.9.3
| Relative Rotation Rates
| 198
|
| 5.10
| Additional Properties
| 199
|
| 5.10.1
| Period as Linking Number
| 199
|
| 5.10.2
| EBK--like Expression for Periods
| 199
|
| 5.10.3
| Poincaré Section
| 201
|
| 5.10.4
| Blow-Up of Branched Manifolds
| 201
|
| 5.10.5
| Branched-Manifold Singularities
| 203
|
| 5.10.6
| Constructing a Branched Manifold from a Map
| 203
|
| 5.10.7
| Topological Entropy
| 203
|
| 5.11
| Subtemplates
| 207
|
| 5.11.1
| Two Alternatives
| 207
|
| 5.11.2
| A Choice
| 210
|
| 5.11.3
| Topological Entropy
| 211
|
| 5.11.4
| Subtemplates of the Smale Horseshoe
| 212
|
| 5.11.5
| Subtemplates Involving Tongues
| 213
|
| 5.12
| Summary
| 215
|
|
| 6
| Topological Analysis Program
| 217
|
| 6.1
| Brief Summary of the Topological Analysis Program
| 217
|
| 6.2
| Overview of the Topological Analysis Program
| 218
|
| 6.2.1
| Find Periodic Orbits
| 218
|
| 6.2.2
| Embed in R3
| 220
|
| 6.2.3
| Compute Topological Invariants
| 220
|
| 6.2.4
| Identify Template
| 221
|
| 6.2.5
| Verify Template
| 222
|
| 6.2.6
| Model Dynamics
| 223
|
| 6.2.7
| Validate Model
| 224
|
| 6.3
| Data
| 225
|
| 6.3.1
| Data Requirements
| 225
|
| 6.3.2
| Processing in the Time Domain
| 226
|
| 6.3.3
| Processing in the Frequency Domain
| 228
|
| 6.4
| Embeddings
| 233
|
| 6.4.1
| Embeddings for Periodically Driven Systems
| 234
|
| 6.4.2
| Differential Embeddings
| 235
|
| 6.4.3
| Differential--Integral Embeddings
| 237
|
| 6.4.4
| Embeddings with Symmetry
| 238
|
| 6.4.5
| Time--Delay Embeddings
| 239
|
| 6.4.6
| Coupled--Oscillator Embeddings
| 241
|
| 6.4.7
| SVD Projections
| 242
|
| 6.4.8
| SVD Embeddings
| 244
|
| 6.4.9
| Embedding Theorems
| 244
|
| 6.5
| Periodic Orbits
| 246
|
| 6.5.1
| Close Returns Plots for Flows
| 246
|
| 6.5.2
| Close Returns in Maps
| 249
|
| 6.5.3
| Metric Methods
| 250
|
| 6.6
| Computation of Topological Invariants
| 251
|
| 6.6.1
| Embed Orbits
| 251
|
| 6.6.2
| Linking Numbers and Relative Rotation Rates
| 252
|
| 6.6.3
| Label Orbits
| 252
|
| 6.7
| Identify Template
| 252
|
| 6.7.1
| Period-1 and Period-2 Orbits
| 252
|
| 6.7.2
| Missing Orbits
| 253
|
| 6.7.3
| More Complicated Branched Manifolds
| 253
|
| 6.8
| Validate Template
| 253
|
| 6.8.1
| Predict Additional Toplogical Invariants
| 254
|
| 6.8.2
| Compare
| 254
|
| 6.8.3
| Global Problem
| 254
|
| 6.9
| Model Dynamics
| 254
|
| 6.10
| Validate Model
| 257
|
| 6.10.1
| Qualitative Validation
| 257
|
| 6.10.2
| Quantitative Validation
| 258
|
| 6.11
| Summary
| 259
|
|
| 7
| Folding Mechanisms: A2
| 261
|
| 7.1
| Belousov--Zhabotinskii Chemical Reaction
| 262
|
| 7.1.1
| Location of Periodic Orbits
| 262
|
| 7.1.2
| Embedding Attempts
| 266
|
| 7.1.3
| Topological Invariants
| 267
|
| 7.1.4
| Template
| 271
|
| 7.1.5
| Dynamical Properties
| 271
|
| 7.1.6
| Models
| 273
|
| 7.1.7
| Model Verification
| 273
|
| 7.2
| Laser with Saturable Absorber
| 275
|
| 7.2.1
| Experimental Setup
| 275
|
| 7.2.2
| Data
| 276
|
| 7.2.3
| Topological Analysis
| 276
|
| 7.2.4
| Useful Observation
| 278
|
| 7.2.5
| Important Conclusion
| 278
|
| 7.3
| Stringed Instrument
| 279
|
| 7.3.1
| Experimental Arrangement
| 279
|
| 7.3.2
| Flow Models
| 280
|
| 7.3.3
| Dynamical Tests
| 281
|
| 7.3.4
| Topological Analysis
| 282
|
| 7.4
| Lasers with Low-Intensity Signals
| 284
|
| 7.4.1
| SVD Embedding
| 286
|
| 7.4.2
| Template Identification
| 286
|
| 7.4.3
| Results of the Analysis
| 288
|
| 7.5
| The Lasers in Lille
| 288
|
| 7.5.1
| Class B Laser Model
| 289
|
| 7.5.2
| CO2 Laser with Modulated Losses
| 295
|
| 7.5.3
| Nd-Doped YAG Laser
| 300
|
| 7.5.4
| Nd-Doped Fiber Laser
| 303
|
| 7.5.5
| Synthesis of Results
| 308
|
| 7.6
| Neuron with Subthreshold Oscillations
| 315
|
| 7.7
| Summary
| 321
|
|
| 8
| Tearing Mechanisms: A3
| 323
|
| 8.1
| Lorenz Equations
| 324
|
| 8.1.1
| Fixed Points
| 325
|
| 8.1.2
| Stability of Fixed Points
| 325
|
| 8.1.3
| Bifurcation Diagram
| 325
|
| 8.1.4
| Templates
| 326
|
| 8.1.5
| Shimizu--Morioka Equations
| 328
|
| 8.2
| Optically Pumped Molecular Laser
| 329
|
| 8.2.1
| Models
| 331
|
| 8.2.2
| Amplitudes
| 332
|
| 8.2.3
| Template
| 333
|
| 8.2.4
| Orbits
| 333
|
| 8.2.5
| Intensities
| 337
|
| 8.3
| Fluid Experiments
| 338
|
| 8.3.1
| Data
| 340
|
| 8.3.2
| Template
| 340
|
| 8.4
| Why A3?
| 341
|
| 8.5
| Summary
| 341
|
|
| 9
| Unfoldings
| 343
|
| 9.1
| Catastrophe Theory as a Model
| 344
|
| 9.1.1
| Overview
| 344
|
| 9.1.2
| Example
| 344
|
| 9.1.3
| Reduction to a Germ
| 346
|
| 9.1.4
| Unfolding the Germ
| 348
|
| 9.1.5
| Summary of Concepts
| 348
|
| 9.2
| Unfolding of Branched Manifolds: Branched Manifolds as Germs
| 348
|
| 9.2.1
| Unfolding of Folds
| 349
|
| 9.2.2
| Unfolding of Tears
| 350
|
| 9.3
| Unfolding within Branched Manifolds: Unfolding of the Horseshoe
| 351
|
| 9.3.1
| Topology of Forcing: Maps
| 352
|
| 9.3.2
| Topology of Forcing: Flows
| 352
|
| 9.3.3
| Forcing Diagrams
| 355
|
| 9.3.4
| Basis Sets of Orbits
| 361
|
| 9.3.5
| Coexisting Basins
| 362
|
| 9.4
| Missing Orbits
| 362
|
| 9.5
| Routes to Chaos
| 363
|
| 9.6
| Summary
| 365
|
|
| 10
| Symmetry
| 367
|
| 10.1
| Information Loss and Gain
| 368
|
| 10.1.1
| Information Loss
| 368
|
| 10.1.2
| Exchange of Symmetry
| 368
|
| 10.1.3
| Information Gain
| 368
|
| 10.1.4
| Symmetries of the Standard Systems
| 368
|
| 10.2
| Cover and Image Relations
| 369
|
| 10.2.1
| General Setup
| 369
|
| 10.3
| Rotation Symmetry 1: Images
| 370
|
| 10.3.1
| Image Equations and Flows
| 370
|
| 10.3.2
| Image of Branched Manifolds
| 373
|
| 10.3.3
| Image of Periodic Orbits
| 374
|
| 10.4
| Rotation Symmetry 2: Covers
| 376
|
| 10.4.1
| Topological Index
| 376
|
| 10.4.2
| Covers of Branched Manifolds
| 378
|
| 10.4.3
| Covers of Periodic Orbits
| 380
|
| 10.5
| Peeling: A New Global Bifurcation
| 380
|
| 10.5.1
| Orbit Perestroika
| 381
|
| 10.5.2
| Covering Equations
| 382
|
| 10.6
| Inversion Symmetry: Driven Oscillators
| 383
|
| 10.6.1
| Periodically Driven Nonlinear Oscillator
| 384
|
| 10.6.2
| Embedding in
M3 subset of R4
| 384
|
| 10.6.3
| Symmetry Reduction
| 385
|
| 10.6.4
| Image Dynamics
| 385
|
| 10.7
| Duffing Oscillator
| 386
|
| 10.8
| van der Pol Oscillator
| 389
|
| 10.9
| Summary
| 395
|
|
| 11
| Flows in Higher Dimensions
| 397
|
| 11.1
| Review of Classification Theory in R3
| 397
|
| 11.2
| General Setup
| 399
|
| 11.2.1
| Spectrum of Lyapunov Exponents
| 400
|
| 11.2.2
| Double Projection
| 400
|
| 11.3
| Flows in R4
| 402
|
| 11.3.1
| Cyclic Phase Spaces
| 402
|
| 11.3.2
| Floppiness and Rigidity
| 402
|
| 11.3.3
| Singularities in Return Maps
| 404
|
| 11.4
| Cusp Bifurcation Diagrams
| 406
|
| 11.4.1
| Cusp Return Maps
| 408
|
| 11.4.2
| Structure in the Control Plane
| 408
|
| 11.4.3
| Comparison with the Fold
| 409
|
| 11.5
| Nonlocal Singularities
| 411
|
| 11.5.1
| Multiple Cusps
| 411
|
| 11.5.2
| Cusps and Folds
| 413
|
| 11.6
| Global Boundary Conditions
| 414
|
| 11.6.1
| R1 and
S1 in Three-Dimensional Flows
| 415
|
| 11.6.2
| Compact Connected Two-Dimensional Domains
| 415
|
| 11.6.3
| Singularities in These Domains
| 416
|
| 11.6.4
| Compact Connected Two-Dimensional Domains
| 416
|
| 11.7
| Summary
| 418
|
|
| 12
| Program for Dynamical Systems Theory
| 421
|
| 12.1
| Reduction of Dimension
| 422
|
| 12.1.1
| Absorbing Manifold
| 424
|
| 12.1.2
| Inertial Manifold
| 424
|
| 12.1.3
| Branched Manifolds
| 424
|
| 12.2
| Equivalence
| 425
|
| 12.2.1
| Diffeomorphisms
| 425
|
| 12.3
| Structure Theory
| 426
|
| 12.3.1
| Reducibility of Dynamical Systems
| 426
|
| 12.4
| Germs
| 427
|
| 12.4.1
| Branched Manifolds
| 427
|
| 12.4.2
| Singular Return Maps
| 427
|
| 12.5
| Unfolding
| 428
|
| 12.6
| Paths
| 430
|
| 12.6.1
| Routes to Chaos
| 430
|
| 12.7
| Rank
| 431
|
| 12.7.1
| Stretching and Squeezing
| 431
|
| 12.8
| Complex Extensions
| 432
|
| 12.8.1
| Fixed-Point Distributions
| 432
|
| 12.8.2
| Singular Return Maps
| 432
|
| 12.9
| Coxeter--Dynkin Diagrams
| 433
|
| 12.9.1
| Fixed-Point Distributions
| 433
|
| 12.9.2
| Singular Return Maps
| 433
|
| 12.10
| Real Forms
| 434
|
| 12.10.1
| Stability of Fixed Points
| 434
|
| 12.10.2
| Singular Return Maps
| 435
|
| 12.11
| Local vs. Global Classification
| 436
|
| 12.11.1
| Nonlocal Folds
| 436
|
| 12.11.2
| Nonlocal Cusps
| 436
|
| 12.12
| Cover--Image Relations
| 437
|
| 12.13
| Symmetry Breaking and Restoration
| 437
|
| 12.13.1
| Entrainment and Synchronization
| 437
|
| 12.14
| Summary
| 439
|
| A.1
| The Fundamental Problem
| 441
|
| A.2
| From Template Matrices to Topological Invariants
| 443
|
| A.2.1
| Classification of Periodic Orbits by Symbolic Names
| 443
|
| A.2.2
| Algebraic Description of a Template
| 444
|
| A.2.3
| Local Torsion
| 445
|
| A.2.4
| Relative Rotation Rates: Examples
| 446
|
| A.2.5
| Relative Rotation Rates: General Case
| 448
|
| A.3
| Identifying Templates from Invariants
| 452
|
| A.3.1
| Using an Independent Symbolic Coding
| 452
|
| A.3.2
| Simultaneous Determination of Symbolic Names and Template
| 455
|
| A.4
| Constructing Generating Partitions
| 459
|
| A.4.1
| Symbolic Encoding as an Interpolation Process
| 459
|
| A.4.2
| Generating partitions for Experimental Data
| 463
|
| A.4.3
| Comparison with Methods Based on Homoclinic Tangencies
| 464
|
| A.4.4
| Symbolic Dynamics on Three Symbols
| 466
|
| A.5
| Summary
| 467
|
|
| References
| 469
|
|
| Topic Index
| 483
|