A new approach to understanding nonlinear dynamics and strange attractorsThe behavior of a physical system may appear irregular or chaotic even when it is completely deterministic and predictable for short periods of time into the future. How does one model the dynamics of a system operating in a chaotic regime? Older tools such as estimates of the spectrum of Lyapunov exponents and estimates of the spectrum of fractal dimensions do not sufficiently answer this question. In a significant evolution of the field of Nonlinear Dynamics, The Topology of Chaos responds to the fundamental challenge of chaotic systems by introducing a new analysis method-Topological Analysis-which can be used to extract, from chaotic data, the topological signatures that determine the stretching and squeezing mechanisms which act on flows in phase space and are responsible for generating chaotic data. Beginning with an example of a laser that has been operated under conditions in which it behaved chaotically, the authors convey the methodology of Topological Analysis through detailed chapters on:* Discrete Dynamical Systems: Maps* Continuous Dynamical Systems: Flows* Topological Invariants* Branched Manifolds* The Topological Analysis Program* Fold Mechanisms* Tearing Mechanisms* Unfoldings* Symmetry* Flows in Higher Dimensions* A Program for Dynamical Systems TheorySuitable at the present time for analyzing "strange attractors" that can be embedded in three-dimensional spaces, this groundbreaking approach offers researchers and practitioners in the discipline a complete and satisfying resolution to the fundamental questions of chaotic systems.
Author(s): Robert Gilmore, Marc Lefranc
Edition: 1
Year: 2002
Language: English
Pages: 518
The Topology of Chaos: Alice in Stretch and Squeezeland......Page 2
Contents......Page 14
Preface......Page 8
1 Introduction......Page 28
1.1 Laser with Modulated Losses......Page 29
1.2 Objectives of a New Analysis Procedure......Page 37
1.3 Preview of Results......Page 38
1.4 Organization of This Work......Page 39
2.1 Introduction......Page 44
2.2 Logistic Map......Page 46
2.3 Bifurcation Diagrams......Page 48
2.4.1 Saddle-Node Bifurcation......Page 50
2.4.2 Period-Doubling Bifurcation......Page 54
2.5.1 Changes of Coordinates......Page 57
2.5.2 Invariants of Conjugacy......Page 58
2.6 Fully Developed Chaos in the Logistic Map......Page 59
2.6.1 Iterates of the Tent Map......Page 60
2.6.3 Sensitivity to Initial Conditions and Mixing......Page 62
2.6.4 Chaos and Density of (Unstable) Periodic Orbits......Page 63
2.6.5 Symbolic Coding of Trajectories: First Approach......Page 65
2.7.1 Partitions......Page 67
2.7.2 Symbolic Dynamics of Expansive Maps......Page 70
2.7.3 Grammar of Chaos: First Approach......Page 73
2.7.4 Kneading Theory......Page 76
2.7.5 Bifurcation Diagram of the Logistic Map Revisited......Page 80
2.8.1 Shifts of Finite Type and Topological Markov Chains......Page 84
2.8.2 Periodic Orbits and Topological Entropy of a Markov Chain......Page 86
2.8.3 Markov Partitions......Page 88
2.8.4 Approximation by Markov Chains......Page 89
2.8.5 Zeta Function......Page 90
2.8.6 Dealing with Grammars......Page 91
2.9.1 Permutation of Periodic Points as a Topological Invariant......Page 94
2.9.2 Topological Entropy of a Periodic Orbit......Page 96
2.9.3 Period-3 Implies Chaos and Sarkovskii 's Theorem......Page 98
2.9.5 Permutations and Orbit Forcing......Page 99
2.10.1 Horseshoe Map......Page 101
2.10.2 Symbolic Dynamics of the Invariant Set......Page 102
2.10.3 Dynamical Properties......Page 105
2.10.4 Variations on the Horseshoe Map: Baker Maps......Page 106
2.11.1 A Once-Folding Map......Page 109
2.11.2 Symbolic Dynamics of the Hénon Map......Page 111
2.12.1 A New Global Topology......Page 117
2.12.2 Frequency Locking and Arnol'd Tongues......Page 118
2.12.3 Chaotic Circle Maps and Annulus Maps......Page 121
2.13 Summary......Page 122
3.1 Definition of Dynamical Systems......Page 124
3.2 Existence and Uniqueness Theorem......Page 125
3.3.1 Dufing Equation......Page 126
3.3.2 van der Pol Equation......Page 127
3.3.3 Lorenz Equations......Page 129
3.3.4 Rössler Equations......Page 132
3.3.5 Examples of Nondynamical Systems......Page 133
3.3.6 Additional Observations......Page 136
3.4.2 Examples......Page 139
3.4.3 Structure Theory......Page 141
3.5.1 Dependence on Topology of Phase Space......Page 143
3.5.2 How to Find Fixed Points in Rn......Page 144
3.5.3 Bifurcations of Fixed Points......Page 145
3.5.4 Stability of Fixed Points......Page 147
3.6.1 Locating Periodic Orbits in Rn-1 × S1......Page 148
3.6.2 Bifurcations of Fixed Points......Page 149
3.6.3 Stability of Fixed Points......Page 150
3.7 Flows near Nonsingular Points......Page 151
3.8 Volume Expansion and Contraction......Page 152
3.9 Stretching and Squeezing......Page 153
3.10 The Fundamental Idea......Page 154
3.11 Summary......Page 155
4 Topological Invariants......Page 158
4.1 Stretching and Squeezing Mechanisms......Page 159
4.2.1 Definitions......Page 163
4.2.2 Reidemeister Moves......Page 165
4.2.3 Braids......Page 166
4.2.4 Examples......Page 169
4.2.5 Linking Numbers for the Horseshoe......Page 170
4.2.6 Linking Numbers for the Lorenz Attractor......Page 171
4.2.8 Local Torsion......Page 173
4.2.9 Writhe and Twist......Page 174
4.2.10 Additional Properties......Page 175
4.3 Relative Rotation Rates......Page 176
4.3.1 Definition......Page 177
4.3.2 How to Compute Relative Rotation Rates......Page 178
4.3.3 Horseshoe Mechanism......Page 182
4.4 Relation between Linking Numbers and Relative Rotation Rates......Page 186
4.5.1 Bifurcation Organization......Page 187
4.5.3 Additional Remarks......Page 188
4.6 Summury......Page 191
5 Branched Manifolds......Page 192
5.1.3 The Ph.D. Candidate......Page 193
5.1.4 Important Observation......Page 195
5.3 General Properties of Branched Manifolds......Page 196
5.4.1 Birman-Williams Projection......Page 198
5.4.2 Statement of the Theorem......Page 200
5.5.1 Strongly Contracting Restriction......Page 202
5.6 Examples of Branched Manifolds......Page 203
5.6.1 Smale-Rössler System......Page 204
5.6.2 Lorenz System......Page 206
5.6.3 Duffing System......Page 207
5.6.4 van der Pol System......Page 209
5.7.1 Local Moves......Page 213
5.7.2 Global Moves......Page 214
5.8 Standard Form......Page 217
5.9.1 Kneading Theory......Page 220
5.9.2 Linking Numbers......Page 224
5.9.3 Relative Rotation Rates......Page 225
5.10.2 EBK-like Expression for Periods......Page 226
5.10.4 Blow-Up of Branched Manifolds......Page 228
5.10.7 Topological Entropy......Page 230
5.11.1 Two Alternatives......Page 234
5.11.2 A Choice......Page 237
5.11.3 Topological Entropy......Page 238
5.11.4 Subtemplates of the Smale Horseshoe......Page 239
5.11.5 Subtemplates Involving Tongues......Page 240
5.12 Summary......Page 242
6.1 Brief Summary of the Topological Analysis Program......Page 244
6.2.1 Find Periodic Orbits......Page 245
6.2.3 Compute Topological Invariants......Page 247
6.2.4 Identify Template......Page 248
6.2.5 Verify Template......Page 249
6.2.6 Model Dynamics......Page 250
6.2.7 Validate Model......Page 251
6.3.1 Data Requirements......Page 252
6.3.2 Processing in the Time Domain......Page 253
6.3.3 Processing in the Frequency Domain......Page 255
6.4 Embeddings......Page 260
6.4.1 Embeddings for Periodically Driven Systems......Page 261
6.4.2 Differential Embeddings......Page 262
6.4.3 Diferential-Integral Embeddings......Page 264
6.4.4 Embeddings with Symmetry......Page 265
6.4.5 Time-Delay Embeddings......Page 266
6.4.6 Coupled-Oscillator Embeddings......Page 268
6.4.7 SVD Projections......Page 269
6.4.9 Embedding Theorems......Page 271
6.5.1 Close Returns Plots for Flows......Page 273
6.5.2 Close Returns in Maps......Page 276
6.5.3 Metric Methods......Page 277
6.6.1 Embed Orbits......Page 278
6.7.1 Period-I and Period-2 Orbits......Page 279
6.8 Validate Template......Page 280
6.9 Model Dynamics......Page 281
6.10.1 Qualitative Validation......Page 284
6.10.2 Quantitative Validation......Page 285
6.11 Summary......Page 286
7 Folding Mechanisms: A2......Page 288
7.1.1 Location of Periodic Orbits......Page 289
7.1.2 Embedding Attempts......Page 293
7.1.3 Topological Invariants......Page 294
7.1.5 Dynamical Properties......Page 298
7.1.7 Model Verification......Page 300
7.2.1 Experimental Setup......Page 302
7.2.3 Topological Analysis......Page 303
7.2.5 Important Conclusion......Page 305
7.3.1 Experimental Arrangement......Page 306
7.3.2 Flow Models......Page 307
7.3.3 Dynamical Tests......Page 308
7.3.4 Topological Analysis......Page 309
7. 4 Lasers with Low-Intensity Signals......Page 311
7.4.2 Template Identification......Page 313
7.5 The Lasers in Lille......Page 315
7.5.1 Class B Laser Model......Page 316
7.5.2 CO2 Laser with Modulated Losses......Page 322
7.5.3 Nd-Doped YAG Laser......Page 327
7.5.4 Nd-Doped Fiber Laser......Page 330
7.5.5 Synthesis of Results......Page 335
7.6 Neuron with Subthreshold Oscillations......Page 342
7.7 Summary......Page 348
8 Tearing Mechanisms: A3......Page 350
8.1 Lorenz Equations......Page 351
8.1.3 Bifurcation Diagram......Page 352
8.1.4 Templates......Page 353
8.1.5 Shimizu-Morioka Equations......Page 355
8.2 Optically Pumped Molecular Laser......Page 356
8.2.1 Models......Page 358
8.2.2 Amplitudes......Page 359
8.2.4 Orbits......Page 360
8.2.5 Intensities......Page 364
8.3 Fluid Experiments......Page 365
8.3.2 Template......Page 367
8.5 Summary......Page 368
9 Unfoldings......Page 370
9.1.2 Example......Page 371
9.1.3 Reduction to a Germ......Page 373
9.2 Unfolding of Branched Manifolds: Branched Manifolds as Germs......Page 375
9.2.1 Unfolding of Folds......Page 376
9.2.2 Unfolding of Tears......Page 377
9.3 Unfolding within Branched Manifolds: Unfolding of the Horseshoe......Page 378
9.3.2 Topology of Forcing: Flows......Page 379
9.3.3 Forcing Diagrams......Page 382
9.3.4 Basis Sets of Orbits......Page 388
9.4 Missing Orbits......Page 389
9.5 Routes to Chaos......Page 390
9.6 Summary......Page 392
10 Symmetty......Page 394
10.1.4 Symmetries of the Standard Systems......Page 395
10.2.1 General Setup......Page 396
10.3.1 Image Equations and Flows......Page 397
10.3.2 Image of Branched Manifolds......Page 400
10.3.3 Image of Periodic Orbits......Page 401
10.4.1 Topological Index......Page 403
10.4.2 Covers of Branched Manifolds......Page 405
10.5 Peeling: A New Global Bifurcation......Page 407
10.5.1 Orbit Perestroika......Page 408
10.5.2 Covering Equations......Page 409
10.6 Inversion Symmetry: Driven Oscillators......Page 410
10.6.2 Embedding in M3 C R4......Page 411
10.6.4 Image Dynamics......Page 412
10.7 Duffing Oscillator......Page 413
10.8 van der Pol Oscillator......Page 416
10.9 Summary......Page 422
11.1 Review of Classification Theory in R3......Page 424
11.2 General Setup......Page 426
11.2.2 Double Projection......Page 427
11.3.2 Floppiness and Rigidity......Page 429
11.3.3 Singularities in Return Maps......Page 431
11.4 Cusp Bifurcation Diagrams......Page 433
11.4.2 Structure in the Control Plane......Page 435
11.4.3 Comparison with the Fold......Page 436
11.5.1 Multiple Cusps......Page 438
11.5.2 Cusps and Folds......Page 440
11.6 Global Boundary Conditions......Page 441
11.6.2 Compact Connected Two-Dimensional Domains......Page 442
11.6.4 Compact Connected Two-Dimensional Domains......Page 443
11.7 Summary......Page 445
12 Program for Dynamical Systems Theory......Page 448
12.1 Reduction of Dimension......Page 449
12.1.3 Branched Manifolds......Page 451
12.2.1 Diffeomorphisms......Page 452
12.3.1 Reducibility of Dynamical Systems......Page 453
12.4.2 Singular Return Maps......Page 454
12.5 Unfolding......Page 455
12.6.1 Routes to Chaos......Page 457
12.7.1 Stretching and Squeezing......Page 458
12.8.2 Singular Return Maps......Page 459
12.9.2 Singular Return Maps......Page 460
12.10.1 Stability of Fixed Points......Page 461
12.10.2 Singular Return Maps......Page 462
12.11.2 Nonlocal Cusps......Page 463
12.13.1 Entrainment and Synchronization......Page 464
12.14 Summary......Page 466
A.1 The Fundamental Problem......Page 468
A.2.1 Classification of Periodic Orbits by Symbolic Names......Page 470
A.2.2 Algebraic Description of a Template......Page 471
A.2.3 Local Torsion......Page 472
A.2.4 Relative Rotation Rates: Examples......Page 473
A.2.5 Relative Rotation Rates: General Case......Page 475
A.3.1 Using an Independent Symbolic Coding......Page 479
A.3.2 Simultaneous Determination of Symbolic Names and Template......Page 482
A.4.1 Symbolic Encoding as an Interpolation Process......Page 486
A.4.2 Generating partitions for Experimental Data......Page 490
A.4.3 Comparison with Methods Based on Homoclinic Tangencies......Page 491
A.4.4 Symbolic Dynamics on Three Symbols......Page 493
A.5 Summary......Page 494
References......Page 496
Topic Index......Page 510
