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<p>.</p> <p>Chaos: Classical and QuantumI: Deterministic Chaos</p> <p>Predrag Cvitanovi c Roberto Artuso Ronnie Mainieri Gregor Tanner G abor Vattay</p> <p>printed February 10, 2013 ChaosBook.org version14.2, Feb 10 2013 ChaosBook.org comments to: predrag@nbi.dk</p> <p>ContentsContributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi xv</p> <p>I1</p> <p>Geometry of chaosOverture 1.1 Why ChaosBook? . . . . . 1.2 Chaos ahead . . . . . . . . 1.3 The future as in a mirror . 1.4 A game of pinball . . . . . 1.5 Chaos for cyclists . . . . . 1.6 Change in time . . . . . . 1.7 To statistical mechanics . . 1.8 Chaos: what is it good for? 1.9 What is not in ChaosBook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .</p> <p>13 . 4 . 5 . 6 . 11 . 15 . 21 . 24 . 25 . 28</p> <p>r esum e 28 commentary 30 guide to exercises 33 exercises 34 references 34</p> <p>2</p> <p>Go with the ow 37 2.1 Dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3 Computing trajectories . . . . . . . . . . . . . . . . . . . . . . . 47r esum e 48 commentary 48 exercises 51 references 52</p> <p>3</p> <p>Discrete time dynamics 3.1 Poincar e sections . . . . . . . 3.2 Computing a Poincar e section 3.3 Mappings . . . . . . . . . . . 3.4 Charting the state space . . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>55 . . . 56 . . . 62 . . . 63 . . . 67</p> <p>r esum e 70 commentary 71 exercises 73 references 74</p> <p>4</p> <p>Local stability 4.1 Flows transport neighborhoods 4.2 Linear ows . . . . . . . . . . 4.3 Stability of ows . . . . . . . 4.4 Neighborhood volume . . . . 4.5 Stability of maps . . . . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>75 . . . . . 75 . . . . . 79 . . . . . 84 . . . . . 89 . . . . . 91</p> <p>r esum e 94 commentary 94 exercises 96 references 96</p> <p>ii</p> <p>CONTENTS</p> <p>iii 99 99 104 106 107</p> <p>5</p> <p>Cycle stability 5.1 Stability of periodic orbits . . . 5.2 Floquet multipliers are invariant 5.3 Stability of Poincar e map cycles 5.4 There goes the neighborhood . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>r esum e 108 commentary 109 exercises 110 references 110</p> <p>6</p> <p>Go straight 6.1 Changing coordinates . . . . . . . . . . . . . . 6.2 Rectication of ows . . . . . . . . . . . . . . 6.3 Collinear helium . . . . . . . . . . . . . . . . 6.4 Rectication of maps . . . . . . . . . . . . . . 6.5 Rectication of a periodic orbit . . . . . . . . . 6.6 Cycle Floquet multipliers are metric invariants .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>111 112 113 114 119 120 121</p> <p>r esum e 123 commentary 123 exercises 125 references 125</p> <p>7</p> <p>Hamiltonian dynamics 7.1 Hamiltonian ows . . . . . . . 7.2 Symplectic group . . . . . . . 7.3 Stability of Hamiltonian ows 7.4 Symplectic maps . . . . . . . 7.5 Poincar e invariants . . . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>127 128 130 132 134 137</p> <p>r esum e 138 commentary 139 exercises 142 references 143</p> <p>8</p> <p>Billiards 145 8.1 Billiard dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 145 8.2 Stability of billiards . . . . . . . . . . . . . . . . . . . . . . . . . 147r esum e 150 commentary 150 exercises 151 references 151</p> <p>9</p> <p>World in a mirror 9.1 Discrete symmetries . . . . . . . . . . . . 9.2 Symmetries of solutions . . . . . . . . . 9.3 Relative periodic orbits . . . . . . . . . . 9.4 Dynamics reduced to fundamental domain 9.5 Invariant polynomials . . . . . . . . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>154 155 163 168 169 171</p> <p>r esum e 172 commentary 174 exercises 176 references 177</p> <p>10 Relativity for cyclists 10.1 Continuous symmetries . . . . . 10.2 Symmetries of solutions . . . . 10.3 Stability . . . . . . . . . . . . . 10.4 Reduced state space . . . . . . . 10.5 Method of images: Hilbert bases</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>180 180 189 194 195 201</p> <p>r esum e 204 commentary 206 exercises 210 references 213</p> <p>11 Charting the state space 219 11.1 Qualitative dynamics . . . . . . . . . . . . . . . . . . . . . . . . 220 11.2 Stretch and fold . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 11.3 Temporal ordering: Itineraries . . . . . . . . . . . . . . . . . . . 227</p> <p>CONTENTS</p> <p>iv</p> <p>11.4 Spatial ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 11.5 Kneading theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 11.6 Symbolic dynamics, basic notions . . . . . . . . . . . . . . . . . 235r esum e 238 commentary 239 exercises 241 references 242</p> <p>12 Stretch, fold, prune 12.1 Goin global: stable/unstable manifolds 12.2 Horseshoes . . . . . . . . . . . . . . . 12.3 Symbol plane . . . . . . . . . . . . . . 12.4 Prune danish . . . . . . . . . . . . . . . 12.5 Recoding, symmetries, tilings . . . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>244 245 249 253 256 257</p> <p>r esum e 260 commentary 261 exercises 263 references 264</p> <p>13 Fixed points, and how to get them 13.1 Where are the cycles? . . . . . 13.2 One-dimensional maps . . . . 13.3 Multipoint shooting method . 13.4 Flows . . . . . . . . . . . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>268 269 273 275 277</p> <p>r esum e 281 commentary 282 exercises 284 references 286</p> <p>II</p> <p>Chaos rules</p> <p>288</p> <p>14 Walkabout: Transition graphs 290 14.1 Matrix representations of topological dynamics . . . . . . . . . . 290 14.2 Transition graphs: wander from node to node . . . . . . . . . . . 292 14.3 Transition graphs: stroll from link to link . . . . . . . . . . . . . 295r esum e 299 commentary 299 exercises 301 references 301</p> <p>15 Counting 15.1 How many ways to get there from here? 15.2 Topological trace formula . . . . . . . . 15.3 Determinant of a graph . . . . . . . . . 15.4 Topological zeta function . . . . . . . . 15.5 Innite partitions . . . . . . . . . . . . 15.6 Shadowing . . . . . . . . . . . . . . . 15.7 Counting cycles . . . . . . . . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>303 304 306 309 313 315 317 318</p> <p>r esum e 321 commentary 323 exercises 324 references 327</p> <p>16 Transporting densities 16.1 Measures . . . . . . . . . . . . . . . . 16.2 Perron-Frobenius operator . . . . . . . 16.3 Why not just leave it to a computer? . . 16.4 Invariant measures . . . . . . . . . . . 16.5 Density evolution for innitesimal times 16.6 Liouville operator . . . . . . . . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>329 330 331 334 336 339 341</p> <p>r esum e 343 commentary 344 exercises 345 references 346</p> <p>CONTENTS</p> <p>v 349 349 356 361 363</p> <p>17 Averaging 17.1 Dynamical averaging . . . 17.2 Evolution operators . . . . 17.3 Averaging in open systems 17.4 Lyapunov exponents . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>r esum e 368 commentary 368 exercises 370 references 370</p> <p>18 Trace formulas 373 18.1 A trace formula for maps . . . . . . . . . . . . . . . . . . . . . . 374 18.2 A trace formula for ows . . . . . . . . . . . . . . . . . . . . . . 379 18.3 An asymptotic trace formula . . . . . . . . . . . . . . . . . . . . 382r esum e 383 commentary 384 exercises 384 references 385</p> <p>19 Spectral determinants 19.1 Spectral determinants for maps . . . . . . . . . . . 19.2 Spectral determinant for ows . . . . . . . . . . . 19.3 Dynamical zeta functions . . . . . . . . . . . . . . 19.4 False zeros . . . . . . . . . . . . . . . . . . . . . 19.5 Spectral determinants vs. dynamical zeta functions 19.6 All too many eigenvalues? . . . . . . . . . . . . .r esum e 397 commentary 398 exercises 399 references 400</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>386 386 388 390 394 394 396</p> <p>20 Cycle expansions 20.1 Pseudocycles and shadowing . . . . . . 20.2 Construction of cycle expansions . . . . 20.3 Periodic orbit averaging . . . . . . . . . 20.4 Cycle formulas for dynamical averages . 20.5 Cycle expansions for nite alphabets . . 20.6 Stability ordering of cycle expansions .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>402 403 405 410 412 416 417</p> <p>r esum e 419 commentary 420 exercises 425 references 427</p> <p>21 Discrete factorization 21.1 Preview . . . . . . . . . . . . . . . . . . 21.2 Discrete symmetries . . . . . . . . . . . . 21.3 Dynamics in the fundamental domain . . 21.4 Factorizations of dynamical zeta functions 21.5 C2 factorization . . . . . . . . . . . . . . 21.6 D3 factorization: 3-disk game of pinball .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>. . . . . .</p> <p>428 429 431 432 436 438 439</p> <p>r esum e 441 commentary 442 exercises 442 references 443</p> <p>III Chaos: what to do about it?22 Why cycle? 22.1 Escape rates . . . . . . . . . . . . . . . . . 22.2 Natural measure in terms of periodic orbits 22.3 Correlation functions . . . . . . . . . . . . 22.4 Trace formulas vs. level sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .</p> <p>445447 447 450 451 453</p> <p>r esum e 454 commentary 455 exercises 456 references 457</p> <p>CONTENTS</p> <p>vi 459 460 464 467 469 474 476 478</p> <p>23 Why does it work? 23.1 Linear maps: exact spectra . . . . . . . . . . 23.2 Evolution operator in a matrix representation 23.3 Classical Fredholm theory . . . . . . . . . . 23.4 Analyticity of spectral determinants . . . . . 23.5 Hyperbolic maps . . . . . . . . . . . . . . . 23.6 Physics of eigenvalues and eigenfunctions . . 23.7 Troubles ahead . . . . . . . . . . . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>r esum e 479 commentary 481 exercises 483 references 483</p> <p>24 Intermittency 24.1 Intermittency everywhere . . 24.2 Intermittency for pedestrians 24.3 Intermittency for cyclists . . 24.4 BER zeta functions . . . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>. . . .</p> <p>486 487 489 501 508</p> <p>r esum e 511 commentary 511 exercises 513 references 514</p> <p>25 Deterministic diusion 516 25.1 Diusion in periodic arrays . . . . . . . . . . . . . . . . . . . . . 517 25.2 Diusion induced by chains of 1-dimensional maps . . . . . . . . 521 25.3 Marginal stability and anomalous diusion . . . . . . . . . . . . . 528r esum e 531 commentary 532 exercises 534 references 534</p> <p>26 Turbulence? 26.1 Fluttering ame front . . . . . . . . . 26.2 Innite-dimensional ows: Numerics 26.3 Visualization . . . . . . . . . . . . . 26.4 Equilibria of equilibria . . . . . . . . 26.5 Why does a ame front utter? . . . . 26.6 Intrinsic parametrization . . . . . . . 26.7 Energy budget . . . . . . . . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>. . . . . . .</p> <p>536 537 540 542 543 545 548 548</p> <p>r esum e 551 commentary 552 exercises 552 references 553</p> <p>27 Irrationally winding 27.1 Mode locking . . . . . . . . . . . . . . . . . 27.2 Local theory: Golden mean renormalization 27.3 Global theory: Thermodynamic averaging . . 27.4 Hausdor dimension of irrational windings . 27.5 Thermodynamics of Farey tree: Farey model .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>. . . . .</p> <p>555 556 561 563 565 567</p> <p>r esum e 569 commentary 569 exercises 572 references 573</p> <p>IV</p> <p>The rest is noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .</p> <p>576578 579 580 583 586</p> <p>28 Noise 28.1 Deterministic transport . . . . . . . 28.2 Brownian diusion . . . . . . . . . 28.3 Noisy trajectories: Continuous time 28.4 Noisy maps: Discrete time . . . . .</p> <p>CONTENTS</p> <p>vii</p> <p>28.5 All nonlinear noise is local . . . . . . . . . . . . . . . . . . . . . 588 28.6 Weak noise: Hamiltonian formulation . . . . . . . . . . . . . . . 590r esum e 592 commentary 592 exercises 595 references 596</p> <p>29 Relaxation for cyclists 600 29.1 Fictitious time relaxation . . . . . . . . . . . . . . . . . . . . . . 601 29.2 Discrete iteration relaxation method . . . . . . . . . . . . . . . . 606 29.3 Least...</p>