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Full Description
A highly valued resource for those who wish to move from the introductory and preliminary understandings and the measurement of chaotic behavior to a more sophisticated and precise understanding of chaotic systems. The authors provide a deep understanding of the structure of strange attractors, how they are classified, and how the information required to identify and classify a strange attractor can be extracted from experimental data.
In its first edition, the Topology of Chaos has been a valuable resource for physicist and mathematicians interested in the topological analysis of dynamical systems. Since its publication in 2002, important theoretical and experimental advances have put the topological analysis program on a firmer basis. This second edition includes relevant results and connects the material to other recent developments. Following significant improvements will be included:
* A gentler introduction to the topological analysis of chaotic systems for the non expert which introduces the problems and questions that one commonly encounters when observing a chaotic dynamics and which are well addressed by a topological approach: existence of unstable periodic orbits, bifurcation sequences, multistability etc.
* A new chapter is devoted to bounding tori which are essential for achieving generality as well as for understanding the influence of boundary conditions.
* The new edition also reflects the progress which had been made towards extending topological analysis to higher-dimensional systems by proposing a new formalism where evolving triangulations replace braids.
* There has also been much progress in the understanding of what is a good representation of a chaotic system, and therefore a new chapter is devoted to embeddings.
* The chapter on topological analysis program will be expanded to cover traditional measures of chaos. This will help to connect those readers who are familiar with those measures and tests to the more sophisticated methodologies discussed in detail in this book.
* The addition of the Appendix with both frequently asked and open questions with answers gathers the most essential points readers should keep in mind and guides to corresponding sections in the book. This will be of great help to those who want to selectively dive into the book and its treatments rather than reading it cover to cover.
What makes this book special is its attempt to classify real physical systems (e.g. lasers) using topological techniques applied to real date (e.g. time series). Hence it has become the experimenter?s guidebook to reliable and sophisticated studies of experimental data for comparison with candidate relevant theoretical models, inevitable to physicists, mathematicians, and engineers studying low-dimensional chaotic systems.
Contents
Preface to Second Edition xvii
Preface to the First Edition xix
1 Introduction 1
1.1 Brief Review of Useful Concepts 2
1.2 Laser with Modulated Losses 4
1.3 Objectives of a New Analysis Procedure 11
1.4 Preview of Results 12
1.5 Organization of This Work 14
2 Discrete Dynamical Systems: Maps 19
2.1 Introduction 19
2.2 Logistic Map 20
2.3 Bifurcation Diagrams 22
2.4 Elementary Bifurcations in the Logistic Map 25
2.5 Map Conjugacy 32
2.6 Fully Developed Chaos in the Logistic Map 34
2.7 One-Dimensional Symbolic Dynamics 42
2.8 Shift Dynamical Systems, Markov Partitions, and Entropy 59
2.9 Fingerprints of Periodic Orbits and Orbit Forcing 70
2.10 Two-Dimensional Dynamics: Smale's Horseshoe 77
2.11 Hénon Map 85
2.12 Circle Maps 96
2.13 Annulus Maps 100
2.14 Summary 104
3 Continuous Dynamical Systems: Flows 105
3.1 Definition of Dynamical Systems 105
3.2 Existence and Uniqueness Theorem 106
3.3 Examples of Dynamical Systems 107
3.4 Change of Variables 120
3.5 Fixed Points 125
3.6 Periodic Orbits 131
3.7 Flows Near Nonsingular Points 134
3.8 Volume Expansion and Contraction 136
3.9 Stretching and Squeezing 137
3.10 The Fundamental Idea 138
3.11 Summary 139
4 Topological Invariants 141
4.1 Stretching and Squeezing Mechanisms 141
4.2 Linking Numbers 145
4.3 Relative Rotation Rates 159
4.4 Relation between Linking Numbers and Relative Rotation Rates 169
4.5 Additional Uses of Topological Invariants 170
4.6 Summary 174
5 Branched Manifolds 175
5.1 Closed Loops 175
5.2 What Does This Have to Do with Dynamical Systems? 178
5.3 General Properties of Branched Manifolds 178
5.4 Birman-Williams Theorem 181
5.5 Relaxation of Restrictions 184
5.6 Examples of Branched Manifolds 186
5.7 Uniqueness and Nonuniqueness 194
5.8 Standard Form 200
5.9 Topological Invariants 201
5.10 Additional Properties 207
5.11 Subtemplates 216
5.12 Summary 224
6 Topological Analysis Program 227
6.1 Brief Summary of the Topological Analysis Program 227
6.2 Overview of the Topological Analysis Program 228
6.3 Data 234
6.4 Embeddings 243
6.5 Periodic Orbits 256
6.6 Computation of Topological Invariants 262
6.7 Identify Template 263
6.8 Validate Template 264
6.9 Model Dynamics 265
6.10 Validate Model 268
6.11 Summary 270
7 FoldingMechanisms: A 2 271
7.1 Belousov-Zhabotinskii Chemical Reaction 272
7.2 Laser with Saturable Absorber 285
7.3 Stringed Instrument 288
7.4 Lasers with Low-Intensity Signals 294
7.5 The Lasers in Lille 297
7.6 The Laser in Zaragoza 322
7.7 Neuron with Subthreshold Oscillations 328
7.8 Summary 334
8 TearingMechanisms: A 3 337
8.1 Lorenz Equations 337
8.2 Optically Pumped Molecular Laser 343
8.3 Fluid Experiments 352
8.4 Why A 3 ? 354
8.5 Summary 354
9 Unfoldings 357
9.1 Catastrophe Theory as a Model 357
9.2 Unfolding of Branched Manifolds: Branched Manifolds as Germs 362
9.3 Unfolding within Branched Manifolds: Unfolding of the Horseshoe 365
9.4 Missing Orbits 375
9.5 Routes to Chaos 377
9.6 Orbit Forcing and Topological Entropy: Mathematical Aspects 378
9.7 Topological Measures of Chaos in Experiments 383
9.8 Summary 389
10 Symmetry 391
10.1 Information Loss and Gain 391
10.2 Cover and Image Relations 393
10.3 Rotation Symmetry 1: Images 394
10.4 Rotation Symmetry 2: Covers 400
10.5 Peeling: a New Global Bifurcation 404
10.6 Inversion Symmetry: Driven Oscillators 407
10.7 Duffing Oscillator 409
10.8 Van der Pol Oscillator 413
10.9 Summary 418
11 Bounding Tori 419
11.1 Stretching & Folding vs. Tearing & Squeezing 420
11.2 Inflation 421
11.3 Boundary of Inflation 422
11.4 Index 423
11.5 Projection 424
11.6 Nature of Singularities 426
11.7 Trinions 427
11.8 Poincaré Surface of Section 429
11.9 Construction of Canonical Forms 429
11.10 Perestroikas 432
11.11 Summary 435
12 Representation Theory for Strange Attractors 437
12.1 Embeddings, Representations, Equivalence 438
12.2 Simplest Class of Strange Attractors 439
12.3 Representation Labels 440
12.4 Equivalence of Representations with Increasing Dimension 446
12.5 Genus-g Attractors 450
12.6 Representation Labels 451
12.7 Equivalence in Increasing Dimension 453
12.8 Summary 455
13 Flows in Higher Dimensions 457
13.1 Review of Classification Theory in R 3 457
13.2 General Setup 459
13.3 Flows in R 4 462
13.4 Cusps in Weakly Coupled, Strongly Dissipative Chaotic Systems 466
13.5 Cusp Bifurcation Diagrams 470
13.6 Nonlocal Singularities 475
13.7 Global Boundary Conditions 477
13.8 From Braids to Triangulations: toward a Kinematics in Higher Dimensions 481
13.9 Summary 490
14 Program for Dynamical Systems Theory 493
14.1 Reduction of Dimension 494
14.2 Equivalence 496
14.3 Structure Theory 497
14.4 Germs 498
14.5 Unfolding 500
14.6 Paths 502
14.7 Rank 502
14.8 Complex Extensions 504
14.9 Coxeter-Dynkin Diagrams 504
14.10 Real Forms 506
14.11 Local vs. Global Classification 507
14.12 Cover-Image Relations 508
14.13 Symmetry Breaking and Restoration 508
14.14 Summary 511
Appendix A Determining Templates from Topological Invariants 513
A.1 The Fundamental Problem 513
A.2 From Template Matrices to Topological Invariants 515
A.3 Identifying Templates from Invariants 523
A.4 Constructing Generating Partitions 531
A.5 Summary 539
Appendix B Embeddings 541
B.1 Diffeomorphisms 541
B.2 Mappings of Data 543
B.3 Tests for Embeddings 547
B.4 Tests of Embedding Tests 549
B.5 Geometric Tests for Embeddings 550
B.6 Dynamical Tests for Embeddings 554
B.7 Topological Test for Embeddings 555
B.8 Postmortem on Embedding Tests 557
B.9 Stationarity 562
B.10 Beyond Embeddings 563
B.11 Summary 563
Appendix C Frequently Asked Questions 565
C.1 Is Template Analysis Valid for Non-Hyperbolic Systems? 565
C.2 Can Template Analysis Be Applied to Weakly Dissipative Systems? 566
C.3 What About Higher-Dimensional Systems? 567
References 569
Index 581
