목차 일부
CONTENTS
Preface ... xi
1 Introduction and overview ... 1
1.1 Some history ... 1
1.2 Examples of chaotic behavior ... 2
1.3 Dynamical systems ... 6
1.4 Attractors ... 10
1.5 Sensitive d...
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목차 전체
CONTENTS
Preface ... xi
1 Introduction and overview ... 1
1.1 Some history ... 1
1.2 Examples of chaotic behavior ... 2
1.3 Dynamical systems ... 6
1.4 Attractors ... 10
1.5 Sensitive dependence on initial conditions ... 15
1.6 Delay coordinates ... 19
Problems ... 21
Notes ... 22
2 One - dimensional maps ... 23
2.1 Piecewise linear one - dimensional maps ... 23
2.2 The logistic map ... 31
2.3 General discussion of smooth one-dimensional maps ... 44
2.4 Examples of applications of one-dimensional maps to chaotic systems of higher dimensionality ... 56
Appendix : Some elementary definitions and theorems concerning sets ... 64
Problems ... 66
Notes ... 68
3 Strange attractors and fractal dimension ... 69
3.1 The box - counting dimension ... 69
3.2 The generalized baker's map ... 75
3.3 Measure and the spectrum of <M:MATH ? xmlns ... '"htt
3.4 Dimension spectrum for the generalized baker's map ... 81
3.5 Character of the natural measure for the generalized baker's map ... 83
3.6 The pointwise dimension ... 86
3.7 Implications and determination of fractal dimension in experiments ... 89
3.8 Embedding ... 93
3.9 Fat fractals ... 97
Appendix : Hausdorff dimension ... 100
Problems ... 103
Notes ... 106
4 Dynamical properties of chaotic systems ... 108
4.1 The horseshoe map and symbolic dynamics ... 108
4.2 Linear stability of steady states and periodic orbits ... 115
4.3 Stable and unstable manifolds ... 122
4.4 Lyapunov exponents ... 129
4.5 Entropies ... 138
4.6 Controlling chaos ... 145
Appendix : Gram-Schmidt orthogonalization ... 148
Problems ... 148
Notes ... 150
5 Nonattracting chaotic sets ... 151
5.1 Fractal basin boundaries ... 152
5.2 Final state sensitivity ... 158
5.3 Structure of fractal basin boundaries ... 161
5.4 Chaotic scattering ... 166
5.5 The dynamics of chaotic scattering ... 170
5.6 The dimensions of nonattracting chaotic sets and their stable and unstable manifolds ... 176
Appendix : Derivation of Eqs. (5. 3) ... 179
Problems ... 181
Notes ... 183
6 Quasiperiodicity ... 184
6.1 Frequency spectrum and attractors ... 184
6.2 The circle map ... 190
6.3 N frequency quatiperiodicity with N>2 ... 200
6.4 Strange nonchaotic attractors of quasiperiodically forced systems ... 205
Problems ... 206
Notes ... 206
7 Chaos in Hamiltonian systems ... 208
7.1 Hamiltonian systems ... 208
7.2 Perturbation of integrable systems ... 224
7.3 Chaos and KAM tori in systems describable by two-dimensional Hamiltonian maps ... 235
7.4 Higher - dimensional systems ... 255
7.5 Strongly chaotic systems ... 257
7.6 The succession of increasingly random systems ... 261
Problems ... 262
Notes ... 264
8 Chaotic transitions ... 266
8.1 The period doubling cascade route to chaotic attractors ... 267
8.2 The intermittency transition to a chaotic attractor ... 272
8.3 Crises ... 277
8.4 The Lorenz system : An example of the creation of a chaotic transient ... 291
8.5 Basin boundary metamorphoses ... 294
8.6 Bifurcations to chaotic scattering ... 299
Problems ... 303
Notes ... 304
9 Multifractals ... 305
9.1 The singularity spectrum <M:MATH ? xmlns ... '"htt
9.2 The partition function formalism ... 313
9.3 Lyapunov partition functions ... 316
9.4 Distribution of finite time Lyapunov exponents ... 322
9.5 Unstable periodic orbits and the natural measure ... 326
9.6 Validity of the Lyapunov and periodic orbits partition functions for nonhyperbolic attractors ... 330
Problems ... 332
Notes ... 332
10 Quantum chaos ... 334
10.1 The energy level spectra of chaotic, bounded, time - independent systems ... 336
10.2 Wavefunctions for classically chaotic, bounded, time - independent systems ... 352
10.3 Temporally periodic systems ... 354
10.4 Quantum chaotic scattering ... 360
Problems ... 361
Notes ... 361
References ... 363
Index ... 382
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