.

0 (hodnocen0 x )

(2) Půjčeno:2x 

BK

2nd. ed.

Oxford : Oxford University, c2000

xxi,650 s.,front. : il.

ISBN 0-19-850723-2 (brož.)

Obsahuje grafy, bibliografické citace, předmluvy, rejstřík

Bibliografie: s. 605-642

Chaos - pojednání

000013088

Contents // First Edition Preface v // First Edition Acknowledgments xi // Second Edition Preface xiii // Second Edition Acknowledgments xv // L 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 8 // 1.4 A Mathematical Model of Biological Population Growth 17 // 1.5 A Model of Convecting Fluids: The Lorenz Model 27 // 1.6 Determinism, Unpredictability, and Divergence of Trajectories 37 // 1.7 Summary and Conclusions 39 // 1.8 Further Reading 40 // 2 The Universality of Chaos 47 // 2.1 Introduction 47 // 2.2 The Feigenbaum Numbers 47 // 2.3 Convergence Ratio for Real Systems 51 // 2.4 Using <5 to Make Predictions 53 // 2.5 Feigenbaum Size Scaling 55 // 2.6 Self-Similarity 56 // 2.7 Other Universal Features 57 // 2.8 Models and the Universality of Chaos 58 // 2.9 Computers and Chaos 61 // 2.10 Further Reading 63 // 2.11 Computer Exercises 64 // II. TOWARD A THEORY // OF NONLINEAR DYNAMICS AND CHAOS 69 // 3 Dynamics in State Space: One and Two Dimensions // 3.1 Introduction // 71 // 71 // xviii // Contents // 3.2 State Space 72 // 3.3 Systems Described by First-Order Differential Equations 74 // 3.4 The No-Intersection Theorem 77 // 3.5 Dissipative Systems and Attractors 78 // 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 84 // 3.9 Dissipation Revisited 86 // 3.10 Two-Dimensional State

Space 87 // 3.11 Two-Dimensional State Space: The General Case 91 // 3.12 Dynamics and Complex Characteristic Values 94 // 3.13 Dissipation and the Divergence Theorem 96 // 3.14 The Jacobian Matrix for Characteristic Values 97 // 3.15 Limit Cycles 100 // 3.16 Poincaré Sections and the Stability of Limit Cycles 102 // 3.17 Bifurcation Theory 106 // 3.18 Summary 113 // 3.19 Further Reading 114 // 3.20 Computer Exercises 116 // 4 Three-Dimensional State Space and Chaos 117 // 4.1 Overview 117 // 4.2 Heuristics 118 // 4.3 Routes to Chaos 121 // 4.4 Three-Dimensional Dynamical Systems 123 // 4.5 Fixed Points in Three Dimensions 124 // 4.6 Limit Cycles and Poincaré Sections 128 // 4.7 Quasi-Periodic Behavior 134 // 4.8 The Routes to Chaos I: Period-Doubling 136 // 4.9 The Routes to Chaos II: Quasi-Periodicity 137 // 4.10 The Routes to Chaos III: Intermittency and Crises 138 // 4.11 The Routes to Chaos IV: // Chaotic Transients and Homoclinic Orbits 138 // 4.12 Homoclinic Tangles and Horseshoes 146 // 4.13 Lyapunov Exponents and Chaos 148 // 4.14 Further Reading 154 // 4.15 Computer Exercises 155 // 5 Iterated Maps 157 // 5.1 Introduction 157 // 5.2 Poincaré Sections and Iterated Maps 158 // 5.3 One-Dimensional Iterated Maps 163 // 5.4 Bifurcations in Iterated Maps: Period-Doubling, Chaos, // and Lyapunov Exponents 166 // Contents // xix // 5.5 Qualitative Universal Behavior: The {/-Sequence 173 // 5.6 Feigenbaum Universality 183 // 5.7 Tent Map 185 // 5.8 Shift Maps and Symbolic

Dynamics 188 // 5.9 The Gaussian Map 192 // 5.10 T wo-Dimensional Iterated Maps 197 // 5.11 The Smale Horseshoe Map 199 // 5.12 Summary 204 // 5.13 Further Reading 204 // 5.14 Computer Exercises 207 // 6 Quasi-Periodicity and Chaos 210 // 6.1 Introduction 210 // 6.2 Quasi-Periodicity and Poincaré Sections 212 // 6.3 Quasi-Periodic Route to Chaos 214 // 6.4 Universality in the Quasi-Periodic Route to Chaos 215 // 6.5 Frequency-Locking 217 // 6.6 Winding Numbers 218 // 6.7 Circle Map 219 // 6.8 The Devil’s Staircase and the Farey Tree 227 // 6.9 Continued Fractions and Fibonacci Numbers 231 // 6.10 On to Chaos and Universality 234 // 6.11 Some Applications 240 // 6.12 Further Reading 246 // 6.13 Computer Exercises 249 // 7 Intermittency and Crises 250 // 7.1 Introduction 250 // 7.2 What Is Intermittency? 250 // 7.3 The Cause of Intermittency 252 // 7.4 Quantitative Theory of Intermittency 256 // 7.5 Types of Intermittency and Experimental Observations 259 // 7.6 Crises 260 // 7.7 Some Conclusions 267 // 7.8 Further Reading 268 // 7.9 Computer Exercises 270 // 8 Hamiltonian Systems 272 // 8.1 Introduction 272 // 8.2 Hamilton’s Equations and the Hamiltonian 273 // 8.3 Phase Space 276 // 8.4 Constants of the Motion and Integrable Hamiltonians 279 // 8.5 Nonintegrable Systems, the KAM Theorem, and Period-Doubling 289 // 8.6 The Hénon-Heiles Hamiltonian 296 // XX // Contents // 8.7 The Chirikov Standard Map 303 // 8.8 The Arnold Cat Map 308 // 8.9 The Dissipative Standard Map

309 // 8.10 Applications of Hamiltonian Dynamics 311 // 8.11 Further Reading 313 // 8.12 Computer Exercises 316 // III. MEASURES OF CHAOS 317 // 9 Quantifying Chaos 319 // 9.1 Introduction 319 // 9.2 Time-Series of Dynamical Variables 320 // 9.3 Lyapunov Exponents 323 // 9.4 Universal Scaling of the Lyapunov Exponent 327 // 9.5 Invariant Measure 330 // 9.6 Kolmogorov-Sinai Entropy 335 // 9.7 Fractal Dimension(s) 341 // 9.8 Correlation Dimension and a Computational Case History 354 // 9.9 Comments and Conclusions 368 // 9.10 Further Reading 369 // 9.11 Computer Exercises 374 // 10 Many Dimensions and Multifractals 375 // 10.1 General Comments and Introduction 375 // 10.2 Embedding (Reconstruction) Spaces 376 // 10.3 Practical Considerations for Embedding Calculations 383 // 10.4 Generalized Dimensions and Generalized Correlation Sums 389 // 10.5 Multifractals and the Spectrum of Scaling Indices Да) 393 // 10.6 Generalized Entropy and the g(A) Spectrum 404 // 10.7 Characterizing Chaos via Periodic Orbits 413 // 10.8 *Statistical Mechanical and Thermodynamic Formalism 415 // 10.9 Wavelet Analysis,  -Calculus, and Related Topics 420 // 10.10 Summary 421 // 10.11 Further Reading 422 // 10.12 Computer Exercises 429 // IV. SPECIAL TOPICS 431 // 11 Pattern Formation and Spatiotemporal Chaos 433 // 11.1 Introduction 433 // 11.2 Two-Dimensional Fluid Flow 436 // 11.3 Coupled-Oscillator Models, Cellular Automata, and Networks 442 // Contents // xxi // 11.4 Transport Models 450

11.5 Reaction-Diffiision Systems: A Paradigm for Pattern Formation 460 // 11.6 Diffusion-Limited Aggregation, Dielectric Breakdown, // and Viscous Fingering: Fractals Revisited 471 // 11.7 Self-Organized Criticality: The Physics of Fractals? 477 // 11.8 Summary 479 // 11.9 Further Reading 480 // 11.10 Computer Exercises 489 // Quantum Chaos, The Theory of Complexity, and Other Topics 490 // 12.1 Introduction 490 // 12.2 Quantum Mechanics and Chaos 490 // 12.3 Chaos and Algorithmic Complexity 508 // 12.4 Miscellaneous Topics: Piece-wise Linear Models, Time-Delay Models, Information Theory, Stochastic Resonance, Computer Networks, Controlling // and Synchronizing Chaos 510 // 12.5 Roll Your Own: Some Simple Chaos Experiments 517 // 12.6 General Comments and Overview: The Future of Chaos 517 // 12.7 Further Reading 519 // Appendix A: Fourier Power Spectra 533 // Appendix В: Bifurcation Theory 541 // Appendix С: The Lorenz Model 547 // Appendix D: The Research Literature on Chaos 559 // Appendix E: Computer Programs 560 // Appendix F: Theory of the Universal Feigenbaum Numbers 568 // Appendix G: The Duffing Double-Well Oscillator 579 // Appendix H: Other Universal Features for // One-Dimensional Iterated Maps 584 // Appendix I: The van der Pol Oscillator 589 // Appendix J: Simple Laser Dynamics Models 598 // References 605 // Index 643