CONTENTS
Preface = ⅴ
Acknowlegments = xi
Ⅰ. THE PHENOMENOLOGY OF CHAOS = 1
1 Three Chaotic Systems = 3
1.1 Prelude = 3
1.2 Linear and Nonlinear Systems = 4
1.3 A Nonlinear Electrical System = 9
1.4 A Mathematical Model of Biological Population Growth = 19
1.5 A Model of Convecting Fluids : The Lorenz Model = 28
1.6 Determinism, Unpredictability, and Divergence of Trajectories = 39
1.7 Summary and Conclusions = 41
1.8 Further Reading = 42
2 The Universality of Chaos = 46
2.1 Introduction = 46
2.2 The Feigenbaum Numbers = 46
2.3 Convergence Ratio for Real Systems = 50
2.4 Using $$\delta $$ to Make Predictions = 52
2.5 Feigenbaum Size Scaling = 54
2.6 Self - Similarity = 56
2.7 Other Universal Features = 58
2.8 Comments on Models and the Universality or Chaos = 58
2.9 Computers and Chaos = 61
2.10 Further Reading = 62
2.11 Computer Exercises = 63
Ⅱ. TOWARD A THEORY OF NONLINEAR DYNAMICS AND CHAOS = 67
3 Dynamics in State Space : One and Two Dimensions = 69
3.1 Introduction = 69
3.2 State Space = 70
3.3 Systems Described by First-Order Differential Equations = 73
3.4 The No-Intersection Theorem = 76
3.5 Dissipative Systems = 77
3.6 One-Dimensional State Space = 79
3.7 Taylor Series Linearization Near Fixed Points = 83
3.8 Trajectories in a One-Dimensional State Space = 85
3.9 Dissipation Revisited = 87
3.10 Two-Dimensional State Space = 88
3.11 Two-Dimensional State Space : The General Case = 93
3.12 Dynamics and Complex Characteristic Values = 96
3.13 Dissipation and the Divergence Theorem = 98
3.14 The Jacobian Matrix for Characteristic Values = 100
3.15 Limit Cycles = 103
3.16 Poicar$${e'}$$ Sections and the Stability of Limit Cycles = 105
3.17 The van der Pol Oscillator = 109
3.18 Bifurcation Theory = 118
3.19 Example - A Simple Laser Model = 126
3.20 Summary = 133
3.21 Further Reading = 134
3.22 Computer Exercises = 135
4 Three - Dimensional State Space and Chaos = 136
4.1 Overview = 136
4.2 Heuristics = 137
4.3 Routes to Chaos = 140
4.4 Three - dimensional Dynamical Systems = 142
4.5 Fixed Points in Three Dimensions = 144
4.6 Limit Cycles and Poincar?? Sections = 148
4.7 Quasi - Periodic Behavior = 154
4.8 The Routes to Chaos Ⅰ : Period-Doubling = 157
4.9 The Routes to Chaos Ⅱ : Quasi-Periodicity = 158
4.10 Routes to Chaos Ⅲ : Intermittency and Crises = 159
4.11 The Routes to Chaos Ⅳ : Chaotic Transients and Homoclinic Orbits = 160
4.12 Homoclinic Tangles and Horseshoes = 169
4.13 Lyapunov Exponents and Chaos = 171
4.14 Further Reading = 178
4.15 Computer Exercises = 179
5 Iterated Maps = 180
5.1 Introduction = 180
5.2 Poincar$${e'}$$ Sections and Iterated Maps = 181
5.3 One - Dimensional Iterated Maps = 187
5.4 Bifurcations in Iterated Maps : Period Doubling, Chaos, and Lyapunov Exponents = 191
5.5 Qualitative Universal Behavior : The U - Sequence = 198
5.6 Theory of the Universal Feigenbaum Number ? = 210
5.7 Derivation of the Feigenbaum Number = 217
5.8 Other Universal Features = 221
5.9 Tent Map = 225
5.10 Shift Maps and Symbolic Dynamics = 228
5.11 The Gaussian Map = 234
5.12 Two-Dimensional Iterated Maps = 239
5.13 The Smale Horseshoe Map = 242
5.14 Summary = 247
5.15 Further Reading = 247
5.16 Computer Exercises = 249
6 Quasi-Periodicity and Chaos = 252
6.1 Introduction = 252
6.2 Quasi-Periodicity and Poincar?? Sections = 254
6.3 Quasi-Periodic Route to Chaos = 256
6.4 Universality in the Quasi-Periodic Route to Chaos = 258
6.5 Frequency-Locking = 260
6.6 Winding Numbers = 261
6.7 Circle Map = 263
6.8 The Devil's Staircase and the Farey Tree = 272
6.9 Continued Fractions and Fibonacci Numbers = 276
6.10 On to Chaos and Universality = 280
6.11 Some Applications = 285
6.12 Further Reading = 292
6.13 Computer Exercises = 294
7 Intermittency and Crises = 295
7.1 Introduction = 295
7.2 What Is Intermittency? = 295
7.3 The Cause of Intermittency = 297
7.4 Quantitative Theory of Intermittency = 301
7.5 Types of Intermittency and Experimental Observations = 304
7.6 Crises = 306
7.7 Some Conclusions = 313
7.8 Further Reading = 313
7.9 Computer Exercises = 315
8 Hamiltonian Systems = 316
8.1 Introduction = 316
8.2 Hamilton's Equations and the Hamiltonian = 318
8.3 Phase Space = 320
8.4 Constants of the Motion and Integrable Hamiltonians = 325
8.5 Nonintegrable Systems, the KAM Theorem, and Period-Doubling = 335
8.6 The H$${e'}$$non - Heiles Hamiltonian = 343
8.7 The Chirikov Standard Map = 351
8.8 The Arnold Cat Map = 356
8.9 The Dissipative Standard Map = 358
8.10 Applications of Hamiltonian Dynamics = 360
8.11 Further Reading = 361
8.12 Computer Exercises = 363
Ⅲ. MEASURES OF CHAOS = 365
9 Quantifying Chaos = 367
9.1 Introduction = 367
9.2 Time - Series of Dynamical Variables = 368
9.3 Lyapunov Exponents = 371
9.4 Universal Scaling of the Lyapunov Exponent = 376
9.5 Invariant Measure = 380
9.6 Kolmogorov-Sinai Entropy = 386
9.7 Fractal Dimension(s) = 392
9.8 Correlation Dimension and a Computational Case History = 407
9.9 Comments and Conclusions = 421
9.10 Further Reading = 421
9.11 Computer Exercises = 426
10 Many Dimensions and Multifractals = 427
10.1 General Comments and Introduction = 427
10.2 Embedding Spaces = 428
10.3 Practical Considerations for Embedding Calculations = 436
10.4 Generalized Dimensions and Generalized Correlation Sums = 443
10.5 Multifractals and the Spectrum of Scaling Indices f(α) = 448
10.6 Generalized Entropy and the g(Λ) Spectrum = 462
10.7 Characterizing Chaos via Periodic Orbits = 472
10.8 Statistical Mechanical and Thermodynamic Formalism = 475
10.9 Summary = 481
10.10 Further Reading = 482
10.11 Computer Exercises = 485
Ⅳ. SPECIAL TOPICS = 487
11 Pattern Formation and Spatiotemporal Chaos = 489
11.1 Introduction = 489
11.2 Two - Dimensional Fluid Flow = 491
11.3 Coupled-Oscillator Models and Cellular Automata = 499
11.4 Transport Models = 507
11.5 Reaction-Diffusion Systems : A Paradigm for Pattern Formation = 520
11.6 Diffusion-Limited Aggregation, Dielectric Break - down, and Viscous Fingering : Fractals Revisited = 533
11.7 Self-Organized Criticality : The Physics of Fractals = 541
11.8 Summary = 542
11.9 Further Reading = 543
11.10 Computer Exercises = 548
12 Quantum Chaos, The Theory of Complexity, and Other Topics = 549
12.1 Introduction = 549
12.2 Quantum Mechanics and Chaos = 549
12.3 Chaos and Algorithmic Complexity = 569
12.4 Miscellaneous Topics : Piece-wise Linear Models, Time-Delay Models, Information Theory, Computer Networks, and Controlling Chaos = 572
12.5 Roll Your Own : Some Simple Chaos Experiments = 578
12.6 General Comments and Overview : The Future of Chaos = 579
12.7 Further Reading = 581
Appendix A : Fourier Power Spectra = 589
Appendix B : Bifurcation Theory = 599
Appendix C : The Lorenz Model = 605
Appendix D : The Research Literature on Chaos = 619
Appendix E : Computer Programs = 620
References = 628
Index = 649
