I Introduction .................................................. 1
1 Chaos everywhere ............................................. 3
1.1 Tilt-A-Whirl ............................................ 4
1.2 Digits of π ............................................. 6
1.3 Butterfly effect ........................................ 9
1.4 Weather prediction ..................................... 12
1.5 Inward spiral .......................................... 13
Further reading ............................................. 15
II Dynamics .................................................... 17
2 Galileo Galilei - Birth of a new science .................... 19
2.1 When will we get there? ................................ 20
2.2 Computer animation ..................................... 21
2.3 Acceleration ........................................... 23
2.4 Free-fall .............................................. 24
2.5 Reconstructing the past ................................ 27
2.6 Projectile motion ...................................... 28
Further reading ............................................. 30
3 Isaac Newton - Dynamics perfected ........................... 32
3.1 Equations of motion .................................... 33
3.2 Force laws ............................................. 34
3.3 Calculus ............................................... 38
Further reading ............................................. 42
4 Celestial mechanics - The clockwork universe ................ 43
4.1 Ptolemy ................................................ 43
4.2 Copernicus ............................................. 44
4.3 Brahe and Kepler ....................................... 47
4.4 Universal gravitation .................................. 49
4.5 Circular orbits ........................................ 50
4.6 Elliptical orbits ...................................... 53
4.7 Clockwork universe ..................................... 56
Further reading ............................................. 58
5 The pendulum-Linear and nonlinear ........................... 60
5.1 Rotational motion ...................................... 61
5.2 Torque ................................................. 62
5.3 Pendulum dynamics ...................................... 64
5.4 Quality factor ......................................... 67
5.5 Pendulum clock ......................................... 69
5.6 Frequency .............................................. 72
5.7 Nonlinearity ........................................... 73
5.8 Where's the chaos? ..................................... 74
Further reading ............................................. 75
6 Sychronization - The Josephson effect ....................... 77
6.1 Hysteresis ............................................. 78
6.2 Multistability ......................................... 80
6.3 Synchronization ........................................ 82
6.4 Symmetry breaking ...................................... 85
6.5 Josephson voltage standard ............................. 87
Further reading ............................................. 90
III Random motion .............................................. 93
7 Chaos forgets the past ...................................... 95
7.1 Period doubling ........................................ 97
7.2 Random rotation ........................................ 99
7.3 Statistics ............................................ 102
7.4 Correlation ........................................... 105
7.5 Voltage - standard redux .............................. 109
Further reading ............................................ 112
8 Chaos takes a random walk .................................. 113
8.1 Probability ........................................... 114
8.2 Quincunx .............................................. 117
8.3 Pascal's triangle ..................................... 119
8.4 Diffusion ............................................. 121
8.5 Chaotic walk .......................................... 124
8.6 In search of true randomness .......................... 125
Further reading ............................................ 126
9 Chaos makes noise .......................................... 128
9.1 Beethoven's Fifth ..................................... 128
9.2 Fourier ............................................... 131
9.3 Frequency analysis .................................... 133
9.4 Music to the ear ...................................... 137
9.5 White noise ........................................... 138
9.6 Random or correlated? ................................. 141
Further reading ............................................ 142
IV Sensitive motion ........................................... 143
10 Edward Lorenz - Butterfly effect ........................... 145
10.1 Lorenz equations ...................................... 146
10.2 Exponential growth .................................... 150
10.3 Exponential and logarithmic functions ................. 153
10.4 Liapunov exponent ..................................... 155
10.5 Exponential decay ..................................... 158
10.6 Weather prediction .................................... 160
Further reading ............................................ 164
11 Chaos comes of age ......................................... 165
11.1 Kinds of chaos ........................................ 165
11.2 Maxwell ............................................... 166
11.3 Poincaré .............................................. 167
11.4 Hadamard .............................................. 168
11.5 Borel ................................................. 170
11.6 Birkhoff .............................................. 172
11.7 Chaos sleeps .......................................... 172
11.8 Golden age of chaos ................................... 174
11.9 Ueda .................................................. 175
11.10 What took so long? ................................... 176
Further reading ............................................ 178
12 Tilt-A-Whirl - Chaos at the amusement park ................. 180
12.1 Sellner ............................................... 181
12.2 Mathematical model .................................... 183
12.3 Dynamics .............................................. 186
12.4 Liapunov exponent ..................................... 189
12.5 Computational limit ................................... 190
12.6 Environmental perturbation ............................ 192
12.7 Long-lived chaotic transients ......................... 196
Further reading ............................................ 199
13 Billiard-ball chaos - Atomic disorder ...................... 200
13.1 Joule and energy ...................................... 201
13.2 Carnot and reversibility .............................. 203
13.3 Clausius and entropy .................................. 206
13.4 Kinetic theory of gases ............................... 208
13.5 Boltzmann and entropy ................................. 211
13.6 Chaos and ergodicity .................................. 214
13.7 Stadium billiards ..................................... 217
13.8 Time's arrow .......................................... 220
13.9 Atomic hypothesis ..................................... 222
Further reading ............................................ 223
14 Iterated maps - Chaos made simple .......................... 225
14.1 Simple chaos .......................................... 226
14.2 Liapunov exponent ..................................... 230
14.3 Stretching and folding ................................ 234
14.4 Ulam and von Neumann - Random numbers ................. 234
14.5 Chaos explodes ........................................ 237
14.6 Shift map - Bare chaos ................................ 239
14.7 Origin of randomness .................................. 242
14.8 Mathemagic ............................................ 245
Further reading ............................................ 248
V Topology of motion ......................................... 251
15 State space - Going with the flow .......................... 253
15.1 State space ........................................... 255
15.2 Attracting point ...................................... 257
15.3 Contracting flow ...................................... 259
15.4 Basin of attraction ................................... 260
15.5 Saddle ................................................ 261
15.6 Limit cycle ........................................... 264
15.7 Poincaré-Bendixson theorem ............................ 267
Further reading ............................................ 268
16 Strange attractor .......................................... 270
16.1 Poincaré section ...................................... 271
16.2 Saddle orbit .......................................... 274
16.3 Period doubling ....................................... 278
16.4 Strange attractor ..................................... 278
16.5 Chaotic flow .......................................... 281
16.6 Stretching and folding ................................ 283
Further reading ............................................ 285
17 Fractal geometry ........................................... 286
17.1 Mathematical monster .................................. 286
17.2 Hausdorff - Fractal dimension ......................... 288
17.3 Mandelbrot - Fractal defined .......................... 290
17.4 Physical fractals ..................................... 293
17.5 Fractal attractor ..................................... 294
Further reading ............................................ 297
18 Stephen Smale - Horseshoe map .............................. 298
18.1 Horseshoe map ......................................... 298
18.2 Invariant set ......................................... 301
18.3 Symbolic dynamics ..................................... 304
18.4 3-D flow .............................................. 307
18.5 Structural stability .................................. 308
18.6 Protests and prizes ................................... 308
Further reading ............................................ 310
19 Henri Poincaré - Topological tangle ........................ 312
19.1 Homoclinic point ...................................... 313
19.2 Homoclinic trajectory ................................. 315
19.3 Homoclinic tangle ..................................... 318
19.4 Fixed-point theorem ................................... 320
19.5 Horseshoe ............................................. 321
19.6 Poincaré-Birkhoff-Smale theorem ....................... 322
19.7 Heteroclinic tangle ................................... 325
19.8 Fractal basin boundary ................................ 326
19.9 Robust chaos .......................................... 329
19.10 Paradox lost ......................................... 330
19.11 Stability of the Solar System ........................ 332
Further reading ............................................ 333
VI Conclusion ................................................. 335
20 Chaos goes to work ......................................... 337
20.1 Randomness ............................................ 337
20.2 Prediction ............................................ 338
20.3 Suppressing chaos ..................................... 341
20.4 Hitchhiker's guide to state space ..................... 345
20.5 Space travel .......................................... 348
20.6 Weather modification .................................. 351
20.7 Adaptation ............................................ 353
20.8 Terra incognita ....................................... 357
Further reading ............................................ 358
Bibliography .................................................. 359
Index ......................................................... 367
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