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Classical and Quantum Chaos【Predrag Cvitanovic等】

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Classical and Quantum Chaos

[书名]Classical and Quantum Chaos
[作者]Predrag Cvitanovic等人
[出版社]电子版（commentsto:predrag@nbi.dk）
[关键词]混沌 chaos 动力系统 非线性
[内容简介]见下文    printed August 24,2000
[分类] 数学>非线性科学丛书 or物理
[书籍来源]www.nbi.dk/ChaosBook/
[版本]
[审校]
[光盘] 不含
[ISBN号] 电子版
[是否是扫描版] 否

Classical and Quantum Chaos

Contributors ................................. x
1 Overture 1
1.1 Why this book? ............................. 2
1.2 Chaos ahead .............................. 3
1.3 Agame of pinball ............................ 4
1.4 Periodic orbit theory .......................... 13
1.5 Evolution operators .......................... 18
1.6 From chaos to statistical mechanics .................. 22
1.7 Semiclassical quantization ....................... 23
1.8 Guide to literature ........................... 25
Guide to exercises ............................. 27
Resum′e .................................. 28
Exercises .................................. 32
2Flows 33
2.1 Dynamical systems ........................... 33
2.2 Flows .................................. 37
2.3 Changing coordinates ......................... 41
2.4 Computing trajectories ......................... 44
2.5 Infinite-dimensional flows ....................... 45
Resum′e .................................. 50
Exercises .................................. 52
3Maps 57
3.1 Poincar′e sections ............................ 57
3.2 Constructing a Poincar′e section .................... 60
3.3 H′enon map ............................... 62
3.4 Billiards ................................. 64
Exercises .................................. 69
4 Local stability 73
4.1 Flows transport neighborhoods .................... 73
4.2 Linear flows ............................... 75
4.3 Nonlinear flows ............................. 80
4.4 Hamiltonian flows ........................... 82
i

ii CONTENTS
4.5 Billiards ................................. 83
4.6 Maps ................................... 86
4.7 Cycle stabilities are metric invariants ................. 87
4.8 Going global: Stable/unstable manifolds ............... 91
Resum′e .................................. 92
Exercises .................................. 94
5 Transporting densities 97
5.1 Measures ................................ 97
5.2 Density evolution ............................ 99
5.3 Invariant measures ...........................102
5.4 Koopman, Perron-Frobenius operators ................105
Resum′e ..................................110
Exercises ..................................112
6 Averaging 117
6.1 Dynamical averaging ..........................117
6.2 Evolution operators ..........................124
6.3 Lyapunov exponents ..........................126
Resum′e ..................................131
Exercises ..................................132
7 Trace formulas 135
7.1 Trace of an evolution operator ....................135
7.2 An asymptotic trace formula .....................142
Resum′e ..................................145
Exercises ..................................146
8 Spectral determinants 147
8.1 Spectral determinants for maps ....................148
8.2 Spectral determinant for flows .....................149
8.3 Dynamical zeta functions .......................151
8.4 False zeros ................................155
8.5 More examples of spectral determinants ...............155
8.6 All too many eigenvalues? .......................158
Resum′e ..................................161
Exercises ..................................163
9Why does it work? 169
9.1 The simplest of spectral determinants: Asingle fixed point ....170
9.2 Analyticity of spectral determinants .................173
9.3 Hyperbolic maps ............................181
9.4 Physics of eigenvalues and eigenfunctions ..............185
9.5 Why not just run it on a computer? .................188
Resum′e ..................................192
Exercises ..................................194

CONTENTS iii
10 Qualitative dynamics 197
10.1 Temporal ordering: Itineraries .....................198
10.2 Symbolic dynamics, basic notions ...................200
10.3 3-disk symbolic dynamics .......................204
10.4 Spatial ordering of “stretch & fold” flows ..............206
10.5 Unimodal map symbolic dynamics ..................210
10.6 Spatial ordering: Symbol square ...................215
10.7 Pruning .................................220
10.8 Topological dynamics .........................222
Resum′e ..................................230
Exercises ..................................233
11 Counting 239
11.1 Counting itineraries ..........................239
11.2 Topological trace formula .......................241
11.3 Determinant of a graph ........................243