Cvitanovic et al. Classical and quantum chaos book (web version 9.2.3, 2002)(750s)_PNc_.pdf

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Classical and QuantumChaos
Predrag Cvitanovic – Roberto Artuso – Per Dahlqvist – Ronnie Mainieri
– Gregor Tanner – Gabor Vattay – Niall Whelan – Andreas Wirzba
—————————————————————-
version 9.2.3
Feb 26 2002
printed June 19, 2002
comments to: predrag@nbi.dk
www.nbi.dk/ChaosBook/
841645633.001.png
Contents
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
Resume .................................. 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
Resume .................................. 50
Exercises .................................. 52
3Maps 57
3.1 Poincare sections ............................ 57
3.2 Constructing a Poincare section .................... 60
3.3 Henon 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
Resume .................................. 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
Resume ..................................110
Exercises ..................................112
6 Averaging 117
6.1 Dynamical averaging ..........................117
6.2 Evolution operators ..........................124
6.3 Lyapunov exponents ..........................126
Resume ..................................131
Exercises ..................................132
7 Trace formulas 135
7.1 Trace of an evolution operator ....................135
7.2 An asymptotic trace formula .....................142
Resume ..................................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
Resume ..................................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
Resume ..................................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
Resume ..................................230
Exercises ..................................233
11 Counting 239
11.1 Counting itineraries ..........................239
11.2 Topological trace formula .......................241
11.3 Determinant of a graph ........................243
11.4 Topological zeta function .......................247
11.5 Counting cycles .............................249
11.6 Infinite partitions ............................252
11.7 Shadowing ................................255
Resume ..................................257
Exercises ..................................260
12 Fixed points, and how to get them 269
12.1 One-dimensional mappings ......................270
12.2 d -dimensional mappings ........................274
12.3 Flows ..................................275
12.4 Periodic orbits as extremal orbits ...................279
12.5 Stability of cycles for maps ......................283
Exercises ..................................288
13 Cycle expansions 293
13.1 Pseudocycles and shadowing ......................293
13.2 Cycle formulas for dynamical averages ................301
13.3 Cycle expansions for finite alphabets .................304
13.4 Stability ordering of cycle expansions .................305
13.5 Dirichlet series .............................308
Resume ..................................311
Exercises ..................................314
14 Why cycle? 319
14.1 Escape rates ...............................319
14.2 Flow conservation sum rules ......................323
14.3 Correlation functions ..........................325
14.4 Trace formulas vs. level sums .....................326
Resume ..................................329
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CONTENTS
Exercises ..................................331
15 Thermodynamic formalism 333
15.1 Renyi entropies .............................333
15.2 Fractal dimensions ...........................338
Resume ..................................342
Exercises ..................................343
16 Intermittency 347
16.1 Intermittency everywhere .......................348
16.2 Intermittency for beginners ......................352
16.3 General intermittent maps .......................365
16.4 Probabilistic or BER zeta functions ..................371
Resume ..................................376
Exercises ..................................378
17 Discrete symmetries 381
17.1 Preview .................................382
17.2 Discrete symmetries ..........................386
17.3 Dynamics in the fundamental domain ................389
17.4 Factorizations of dynamical zeta functions ..............393
17.5 C 2 factorization .............................395
17.6 C 3 v factorization: 3-disk game of pinball ...............397
Resume ..................................400
Exercises ..................................403
18 Deterministic diffusion 407
18.1 Diffusion in periodic arrays ......................408
18.2 Diffusion induced by chains of 1- d maps ...............412
Resume ..................................421
Exercises ..................................424
19Irrationally winding 425
19.1 Mode locking ..............................426
19.2 Local theory: “Golden mean” renormalization ............433
19.3 Global theory: Thermodynamic averaging ..............435
19.4 Hausdorff dimension of irrational windings ..............436
19.5 Thermodynamics of Farey tree: Farey model ............438
Resume ..................................444
Exercises ..................................447
20 Statistical mechanics 449
20.1 The thermodynamic limit .......................449
20.2 Ising models ...............................452
20.3 Fisher droplet model ..........................455
20.4 Scaling functions ............................461

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