Kautz R. Chaos: the science of predictable random motion (Oxford; New York, 2011). - ОГЛАВЛЕНИЕ / CONTENTS
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ОбложкаKautz R. Chaos: the science of predictable random motion. - Oxford; New York: Oxford University Press, 2011. - xiii, 369 p.: ill. + 1 CD-ROM. - Bibliogr.: p.359-365. - Ind.: p.367-369. - ISBN 978-0-19-959458-0
 

Оглавление / Contents
 
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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