The Topology of Chaos

Overview

"The 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
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Overview

"The 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." Suitable at the present time for analyzing "strange attractors" that can be embedded in three-dimensional spaces, this approach offers researchers and practitioners in the discipline a complete and satisfying resolution to the fundamental questions of chaotic systems.
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Editorial Reviews

From The Critics
Physicists Gilmore (Drexel, U., Philadelphia) and Lefrance (U. of Sciences and Technology, de Lille, France) analyze data generated by a dynamical system operating in a chaotic regime. Specifically, they describe how to extract from chaotic data topological signatures that determine the stretching and squeezing mechanisms that act on flows in phase space and are responsible for generating chaotic data. The topological methods they develop were in response to the challenge of analyzing chaotic data sets generated by a laser operating under conditions in which it behaved chaotically. Annotation c. Book News, Inc., Portland, OR
From the Publisher
"…an abundance of interesting physically relevant examples. The figures are numerous and illustrative." (Dynamical Systems Magazine, January 2006)

"A short review can only hint at the wealth of ideas here...highly recommended." (Choice, Vol. 40, No. 7, March 2003)

"In this third book Gilmore and Lefranc step one more rung up the ladder of dynamical complexity..." (American Journal of Physics, Vol. 71, No. 5, May 2003)

"This authoritative monograph advances innovative methods for the analysis of chaotic systems." (Journal of Mathematical Psychology, Vol. 47, 2003)

"...contains a wealth of material and, in particular, many practical examples of how topological information can be extracted from experimental time series." (Mathematical Reviews, 2003k)

"...well written, with rigorous and clear exposition of the material, and is pleasant to read..." (Zentralblatt Math, 2003)

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Product Details

  • ISBN-13: 9783527410675
  • Publisher: Wiley
  • Publication date: 5/8/2012
  • Language: German
  • Edition description: Second Revised and Enlarged Edition
  • Edition number: 2
  • Pages: 618
  • Sales rank: 1,453,782
  • Product dimensions: 6.70 (w) x 9.50 (h) x 1.30 (d)

Meet the Author

ROBERT GILMORE, PhD, is a professor in the Physics Department of Drexel University, Philadelphia, Pennsylvania.

MARC LEFRANC, PhD, is a researcher at the Centre National de la Recherche Scientifique in the Laboratoire de Physique des Lasers, Atomes, Molecules at the Universite des Sciences et Technologies de Lille, France.

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Table of Contents

Preface
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
2.5 Map Conjugacy 30
2.6 Fully Developed Chaos in the Logistic Map 32
2.7 One-Dimensional Symbolic Dynamics 40
2.8 Shift Dynamical Systems, Markov Partitions, and Entropy 57
2.9 Fingerprints of Periodic Orbits and Orbit Forcing 67
2.10 Two-Dimensional Dynamics: Smale's Horseshoe 74
2.11 Henon Map 82
2.12 Circle Maps 90
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
3.4 Change of Variables 112
3.5 Fixed Points 116
3.6 Periodic Orbits 121
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
4.3 Relative Rotation Rates 149
4.4 Relation between Linking Numbers and Relative Rotation Rates 159
4.5 Additional Uses of Topological Invariants 160
4.6 Summary 164
5 Branched Manifolds 165
5.1 Closed Loops 166
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
5.5 Relaxation of Restrictions 175
5.6 Examples of Branched Manifolds 176
5.7 Uniqueness and Nonuniqueness 186
5.8 Standard Form 190
5.9 Topological Invariants 193
5.10 Additional Properties 199
5.11 Subtemplates 207
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
6.3 Data 225
6.4 Embeddings 233
6.5 Periodic Orbits 246
6.6 Computation of Topological Invariants 251
6.7 Identify Template 252
6.8 Validate Template 253
6.9 Model Dynamics 254
6.10 Validate Model 257
6.11 Summary 259
7 Folding Mechanisms: A[subscript 2] 261
7.1 Belousov-Zhabotinskii Chemical Reaction 262
7.2 Laser with Saturable Absorber 275
7.3 Stringed Instrument 279
7.4 Lasers with Low-Intensity Signals 284
7.5 The Lasers in Lille 288
7.6 Neuron with Subthreshold Oscillations 315
7.7 Summary 321
8 Tearing Mechanisms: A[subscript 3] 323
8.1 Lorenz Equations 324
8.2 Optically Pumped Molecular Laser 329
8.3 Fluid Experiments 338
8.4 Why A[subscript 3]? 341
8.5 Summary 341
9 Unfoldings 343
9.1 Catastrophe Theory as a Model 344
9.2 Unfolding of Branched Manifolds: Branched Manifolds as Germs 348
9.3 Unfolding within Branched Manifolds: Unfolding of the Horseshoe 351
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
10.2 Cover and Image Relations 369
10.3 Rotation Symmetry 1: Images 370
10.4 Rotation Symmetry 2: Covers 376
10.5 Peeling: A New Global Bifurcation 380
10.6 Inversion Symmetry: Driven Oscillators 383
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 R[superscript 3] 397
11.2 General Setup 399
11.3 Flows in R[superscript 4] 402
11.4 Cusp Bifurcation Diagrams 406
11.5 Nonlocal Singularities 411
11.6 Global Boundary Conditions 414
11.7 Summary 418
12 Program for Dynamical Systems Theory 421
12.1 Reduction of Dimension 422
12.2 Equivalence 425
12.3 Structure Theory 426
12.4 Germs 427
12.5 Unfolding 428
12.6 Paths 430
12.7 Rank 431
12.8 Complex Extensions 432
12.9 Coxeter-Dynkin Diagrams 433
12.10 Real Forms 434
12.11 Local vs. Global Classification 436
12.12 Cover-Image Relations 437
12.13 Symmetry Breaking and Restoration 437
12.14 Summary 439
App. A Determining Templates from Topological Invariants 441
References 469
Topic Index 483
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