Chaos-Classical and quantum

. Chaos: Classical and QuantumI: Deterministic Chaos

Predrag Cvitanovic Roberto Artuso Ronnie Mainieri Gregor Tanner Gabor Vattay

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ChaosBook.org version15.6, Mar 15 2015 printed March 25, 2015ChaosBook.org comments to: [email protected]

Contents

0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

I Geometry of chaos 1

1 Overture 31.1 Why ChaosBook? . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Chaos ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 The future as in a mirror . . . . . . . . . . . . . . . . . . . . . . 61.4 A game of pinball . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Chaos for cyclists . . . . . . . . . . . . . . . . . . . . . . . . . . 151.6 Change in time . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.7 To statistical mechanics . . . . . . . . . . . . . . . . . . . . . . . 241.8 Chaos: what is it good for? . . . . . . . . . . . . . . . . . . . . . 251.9 What is not in ChaosBook . . . . . . . . . . . . . . . . . . . . . 28resume 28 commentary 30 guide to exercises 33 exercises 34 references 34

2 Go with the flow 372.1 Dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . 372.2 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.3 Changing coordinates . . . . . . . . . . . . . . . . . . . . . . . . 462.4 Life in extreme dimensions . . . . . . . . . . . . . . . . . . . . . 472.5 Computing trajectories . . . . . . . . . . . . . . . . . . . . . . . 52resume 53commentary 542.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57exercises 60 references 62

3 Discrete time dynamics 663.1 Poincare sections . . . . . . . . . . . . . . . . . . . . . . . . . . 673.2 Computing a Poincare section . . . . . . . . . . . . . . . . . . . 713.3 Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73resume 75 commentary 763.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78exercises 81 references 82

ii

CONTENTS iii

4 Local stability 844.1 Flows transport neighborhoods . . . . . . . . . . . . . . . . . . . 844.2 Computing the Jacobian matrix . . . . . . . . . . . . . . . . . . . 884.3 A linear diversion . . . . . . . . . . . . . . . . . . . . . . . . . . 894.4 Stability of flows . . . . . . . . . . . . . . . . . . . . . . . . . . 904.5 Stability of maps . . . . . . . . . . . . . . . . . . . . . . . . . . 914.6 Stability of Poincare return maps . . . . . . . . . . . . . . . . . . 924.7 Neighborhood volume . . . . . . . . . . . . . . . . . . . . . . . 94resume 95 commentary 964.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97exercises 103 references 104

5 Cycle stability 1075.1 Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.2 Periodic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.3 Floquet multipliers are invariant . . . . . . . . . . . . . . . . . . 1125.4 Floquet multipliers are metric invariants . . . . . . . . . . . . . . 1145.5 Stability of Poincare map cycles . . . . . . . . . . . . . . . . . . 1155.6 There goes the neighborhood . . . . . . . . . . . . . . . . . . . . 116resume 117 commentary 1175.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118exercises 120 references 120

6 Lyapunov exponents 1216.1 Stretch, strain and twirl . . . . . . . . . . . . . . . . . . . . . . . 1226.2 Lyapunov exponents . . . . . . . . . . . . . . . . . . . . . . . . 123resume 126 commentary 1266.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129exercises 129 references 130

7 Fixed points 1337.1 One-dimensional maps . . . . . . . . . . . . . . . . . . . . . . . 1347.2 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136resume 137 commentary 1387.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138exercises 141 references 141

8 Hamiltonian dynamics 1448.1 Hamiltonian flows . . . . . . . . . . . . . . . . . . . . . . . . . . 1458.2 Symplectic group . . . . . . . . . . . . . . . . . . . . . . . . . . 1478.3 Stability of Hamiltonian flows . . . . . . . . . . . . . . . . . . . 1498.4 Symplectic maps . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.5 Poincare invariants . . . . . . . . . . . . . . . . . . . . . . . . . 154resume 155 commentary 156 exercises 159 references 160

9 Billiards 1629.1 Billiard dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 1629.2 Stability of billiards . . . . . . . . . . . . . . . . . . . . . . . . . 164resume 167 commentary 167 exercises 168 references 169

CONTENTS iv

10 Flips, slides and turns 17110.1 Discrete symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 17110.2 Subgroups, cosets, classes . . . . . . . . . . . . . . . . . . . . . 17410.3 Orbits, quotient space . . . . . . . . . . . . . . . . . . . . . . . . 176resume 178 commentary 17910.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180exercises 182 references 182

11 World in a mirror 18411.1 Symmetries of solutions . . . . . . . . . . . . . . . . . . . . . . 18411.2 Relative periodic orbits . . . . . . . . . . . . . . . . . . . . . . . 18711.3 Dynamics reduced to fundamental domain . . . . . . . . . . . . . 18911.4 Invariant polynomials . . . . . . . . . . . . . . . . . . . . . . . . 190resume 191 commentary 19211.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193exercises 197 references 198

12 Relativity for cyclists 20012.1 Continuous symmetries . . . . . . . . . . . . . . . . . . . . . . . 20112.2 Symmetries of solutions . . . . . . . . . . . . . . . . . . . . . . 20612.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211resume 212 commentary 21312.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214exercises 219 references 219

13 Slice & dice 22213.1 Only dead fish go with the flow: moving frames . . . . . . . . . . 22313.2 Symmetry reduction . . . . . . . . . . . . . . . . . . . . . . . . . 22413.3 Bringing it all back home: method of slices . . . . . . . . . . . . 22513.4 Dynamics within a slice . . . . . . . . . . . . . . . . . . . . . . . 22713.5 First Fourier mode slice . . . . . . . . . . . . . . . . . . . . . . . 22913.6 Stability within a slice . . . . . . . . . . . . . . . . . . . . . . . 23113.7 Method of images: Hilbert bases . . . . . . . . . . . . . . . . . . 233resume 234 commentary 23513.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240exercises 240 references 241

14 Charting the state space 24714.1 Qualitative dynamics . . . . . . . . . . . . . . . . . . . . . . . . 24814.2 Stretch and fold . . . . . . . . . . . . . . . . . . . . . . . . . . . 25114.3 Temporal ordering: Itineraries . . . . . . . . . . . . . . . . . . . 25414.4 Spatial ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . 25614.5 Kneading theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 25714.6 Symbolic dynamics, basic notions . . . . . . . . . . . . . . . . . 261resume 264 commentary 26414.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266exercises 269 references 270

CONTENTS v

15 Stretch, fold, prune 27315.1 Goin global: stable/unstable manifolds . . . . . . . . . . . . . . 27415.2 Horseshoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27815.3 Symbol plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28115.4 Prune danish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28415.5 Recoding, symmetries, tilings . . . . . . . . . . . . . . . . . . . . 28615.6 Charting the state space . . . . . . . . . . . . . . . . . . . . . . . 288resume 291 commentary 29315.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294exercises 298 references 299

16 Fixed points, and how to get them 30416.1 Where are the cycles? . . . . . . . . . . . . . . . . . . . . . . . . 30516.2 Multipoint shooting method . . . . . . . . . . . . . . . . . . . . 30716.3 Cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309resume 311 commentary 31216.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312exercises 314 references 315

II Chaos rules 317

17 Walkabout: Transition graphs 31917.1 Matrix representations of topological dynamics . . . . . . . . . . 31917.2 Transition graphs: wander from node to node . . . . . . . . . . . 32117.3 Transition graphs: stroll from link to link . . . . . . . . . . . . . 323resume 325 commentary 32617.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327exercises 330 references 331

18 Counting 33218.1 How many ways to get there from here? . . . . . . . . . . . . . . 33318.2 Topological trace formula . . . . . . . . . . . . . . . . . . . . . . 33418.3 Determinant of a graph . . . . . . . . . . . . . . . . . . . . . . . 33718.4 Topological zeta function . . . . . . . . . . . . . . . . . . . . . . 33918.5 Infinite partitions . . . . . . . . . . . . . . . . . . . . . . . . . . 34118.6 Shadowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34318.7 Counting cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . 344resume 346 commentary 34818.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349exercises 353 references 356

19 Transporting densities 35819.1 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35919.2 Perron-Frobenius operator . . . . . . . . . . . . . . . . . . . . . 36019.3 Why not just leave it to a computer? . . . . . . . . . . . . . . . . 36219.4 Invariant measures . . . . . . . . . . . . . . . . . . . . . . . . . 36419.5 Density evolution for infinitesimal times . . . . . . . . . . . . . . 36819.6 Liouville operator . . . . . . . . . . . . . . . . . . . . . . . . . . 369

CONTENTS vi

resume 371 commentary 37219.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373exercises 374 references 376

20 Averaging 37820.1 Dynamical averaging . . . . . . . . . . . . . . . . . . . . . . . . 37820.2 Moments, cumulants . . . . . . . . . . . . . . . . . . . . . . . . 38320.3 Evolution operators . . . . . . . . . . . . . . . . . . . . . . . . . 38520.4 Averaging in open systems . . . . . . . . . . . . . . . . . . . . . 39020.5 Evolution operator evaluation of Lyapunov exponents . . . . . . . 391resume 392 commentary 39320.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394exercises 396 references 396

21 Trace formulas 39821.1 A trace formula for maps . . . . . . . . . . . . . . . . . . . . . . 39921.2 A trace formula for flows . . . . . . . . . . . . . . . . . . . . . . 40321.3 An asymptotic trace formula . . . . . . . . . . . . . . . . . . . . 406resume 407 commentary 40821.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408exercises 409 references 410

22 Spectral determinants 41122.1 Spectral determinants for maps . . . . . . . . . . . . . . . . . . . 41122.2 Spectral determinant for flows . . . . . . . . . . . . . . . . . . . 41222.3 Dynamical zeta functions . . . . . . . . . . . . . . . . . . . . . . 41422.4 False zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41722.5 Spectral determinants vs. dynamical zeta functions . . . . . . . . 41822.6 All too many eigenvalues? . . . . . . . . . . . . . . . . . . . . . 418resume 420 commentary 42122.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422exercises 424 references 426

23 Cycle expansions 42823.1 Pseudo-cycles and shadowing . . . . . . . . . . . . . . . . . . . 42923.2 Construction of cycle expansions . . . . . . . . . . . . . . . . . . 43223.3 Periodic orbit averaging . . . . . . . . . . . . . . . . . . . . . . . 43623.4 Flow conservation sum rules . . . . . . . . . . . . . . . . . . . . 43723.5 Cycle formulas for dynamical averages . . . . . . . . . . . . . . . 43823.6 Cycle expansions for finite alphabets . . . . . . . . . . . . . . . . 44223.7 Stability ordering of cycle expansions . . . . . . . . . . . . . . . 443resume 445 commentary 44623.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450exercises 451 references 452

24 Deterministic diusion 45424.1 Diusion in periodic arrays . . . . . . . . . . . . . . . . . . . . . 45524.2 Diusion induced by chains of 1-dimensional maps . . . . . . . . 45924.3 Marginal stability and anomalous diusion . . . . . . . . . . . . . 466

CONTENTS vii

resume 469 commentary 470 exercises 472 references 472

25 Discrete symmetry factorization 47425.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47525.2 Discrete symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 47725.3 Dynamics in the fundamental domain . . . . . . . . . . . . . . . 47825.4 Factorization of spectral determinants . . . . . . . . . . . . . . . 48125.5 C2 = D1 factorization . . . . . . . . . . . . . . . . . . . . . . . . 48325.6 D3 factorization: 3-disk game of pinball . . . . . . . . . . . . . . 485resume 487 commentary 48825.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488exercises 489 references 490

26 Continuous symmetry factorization 49226.1 Compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 49326.2 Continuous symmetries of dynamics . . . . . . . . . . . . . . . . 49926.3 Symmetry reduced trace formula for flows . . . . . . . . . . . . . 501resume 502 commentary 50326.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504exercises 507 references 507

III Chaos: what to do about it? 510

27 Why cycle? 51227.1 Escape rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51227.2 Natural measure in terms of periodic orbits . . . . . . . . . . . . 51527.3 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . 51627.4 Trace formulas vs. level sums . . . . . . . . . . . . . . . . . . . . 518resume 519 commentary 520 exercises 521 references 522

28 Why does it work? 52428.1 Linear maps: exact spectra . . . . . . . . . . . . . . . . . . . . . 52528.2 Evolution operator in a matrix representation . . . . . . . . . . . 52928.3 Classical Fredholm theory . . . . . . . . . . . . . . . . . . . . . 53228.4 Analyticity of spectral determinants . . . . . . . . . . . . . . . . 53428.5 Hyperbolic maps . . . . . . . . . . . . . . . . . . . . . . . . . . 53928.6 Physics of eigenvalues and eigenfunctions . . . . . . . . . . . . . 54128.7 Troubles ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . 543resume 544 commentary 546 exercises 548 references 548

29 Intermittency 55129.1 Intermittency everywhere . . . . . . . . . . . . . . . . . . . . . . 55229.2 Intermittency for pedestrians . . . . . . . . . . . . . . . . . . . . 55429.3 Intermittency for cyclists . . . . . . . . . . . . . . . . . . . . . . 56629.4 BER zeta functions . . . . . . . . . . . . . . . . . . . . . . . . . 573resume 576 commentary 576 exercises 578 references 579

CONTENTS viii

30 Turbulence? 58130.1 Fluttering flame front . . . . . . . . . . . . . . . . . . . . . . . . 58230.2 Infinite-dimensional flows: Numerics . . . . . . . . . . . . . . . 58530.3 Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58630.4 Equilibria of equilibria . . . . . . . . . . . . . . . . . . . . . . . 58730.5 Why does a flame front flutter? . . . . . . . . . . . . . . . . . . . 58930.6 Intrinsic parametrization . . . . . . . . . . . . . . . . . . . . . . 59230.7 Energy budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593resume 596 commentary 596 exercises 597 references 598

31 Irrationally winding 60131.1 Mode locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60231.2 Local theory: Golden mean renormalization . . . . . . . . . . . 60731.3 Global theory: Thermodynamic averaging . . . . . . . . . . . . . 60931.4 Hausdor dimension of irrational windings . . . . . . . . . . . . 61131.5 Thermodynamics of Farey tree: Farey model . . . . . . . . . . . . 613resume 615 commentary 615 exercises 618 references 619

IV The rest is noise 622

32 Noise 62432.1 Deterministic transport . . . . . . . . . . . . . . . . . . . . . . . 62532.2 Brownian diusion . . . . . . . . . . . . . . . . . . . . . . . . . 62632.3 Noisy trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . 62932.4 Noisy maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63332.5 All nonlinear noise is local . . . . . . . . . . . . . . . . . . . . . 63432.6 Weak noise: Hamiltonian formulation . . . . . . . . . . . . . . . 637resume 639 commentary 639 exercises 641 references 644

33 Relaxation for cyclists 64833.1 Fictitious time relaxation . . . . . . . . . . . . . . . . . . . . . . 64933.2 Discrete iteration relaxation method . . . . . . . . . . . . . . . . 65433.3 Least action method . . . . . . . . . . . . . . . . . . . . . . . . . 658resume 658 commentary 659 exercises 661 references 661

V Quantum chaos 66434 Prologue 666

34.1 Quantum pinball . . . . . . . . . . . . . . . . . . . . . . . . . . 66734.2 Quantization of helium . . . . . . . . . . . . . . . . . . . . . . . 669commentary 670 references 671

35 Quantum mechanics- the short short version 672exercises 675

CONTENTS ix

36 WKB quantization 67736.1 WKB ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67736.2 Method of stationary phase . . . . . . . . . . . . . . . . . . . . . 68036.3 WKB quantization . . . . . . . . . . . . . . . . . . . . . . . . . 68136.4 Beyond the quadratic saddle point . . . . . . . . . . . . . . . . . 683resume 684 commentary 685 exercises 686 references 686

37 Semiclassical evolution 68737.1 Hamilton-Jacobi theory . . . . . . . . . . . . . . . . . . . . . . . 68737.2 Semiclassical propagator . . . . . . . . . . . . . . . . . . . . . . 69637.3 Semiclassical Greens function . . . . . . . . . . . . . . . . . . . 699resume 705 commentary 706 exercises 708 references 709

38 Semiclassical quantization 71038.1 Trace formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71038.2 Semiclassical spectral determinant . . . . . . . . . . . . . . . . . 71638.3 One-dof systems . . . . . . . . . . . . . . . . . . . . . . . . . . 71738.4 Two-dof systems . . . . . . . . . . . . . . . . . . . . . . . . . . 718resume 719 commentary 720 exercises 722 references 722

39 Quantum scattering 72439.1 Density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . 72439.2 Quantum mechanical scattering matrix . . . . . . . . . . . . . . . 72839.3 Krein-Friedel-Lloyd formula . . . . . . . . . . . . . . . . . . . . 72939.4 Wigner time delay . . . . . . . . . . . . . . . . . . . . . . . . . . 732commentary 734 exercises 735 references 735

40 Chaotic multiscattering 73840.1 Quantum mechanical scattering matrix . . . . . . . . . . . . . . . 73940.2 N-scatterer spectral determinant . . . . . . . . . . . . . . . . . . 74240.3 Semiclassical 1-disk scattering . . . . . . . . . . . . . . . . . . . 74640.4 From quantum cycle to semiclassical cycle . . . . . . . . . . . . . 75340.5 Heisenberg uncertainty . . . . . . . . . . . . . . . . . . . . . . . 756commentary 756 references 757

41 Helium atom 75841.1 Classical dynamics of collinear helium . . . . . . . . . . . . . . . 75941.2 Chaos, symbolic dynamics and periodic orbits . . . . . . . . . . . 76041.3 Local coordinates, Jacobian matrix . . . . . . . . . . . . . . . . . 76441.4 Getting ready . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76641.5 Semiclassical quantization of collinear helium . . . . . . . . . . . 768resume 775 commentary 775 exercises 777 references 778

42 Diraction distraction 77942.1 Quantum eavesdropping . . . . . . . . . . . . . . . . . . . . . . 77942.2 An application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785resume 790 commentary 791 exercises 792 references 793

Epilogue 795

CONTENTS x

Index 800

CONTENTS xi

Volume www: Appendices on ChaosBook.org

VI Web Appendices 819

A A brief history of chaos 821A.1 Chaos is born . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821A.2 Chaos grows up . . . . . . . . . . . . . . . . . . . . . . . . . . . 825A.3 Chaos with us . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826A.4 Periodic orbit theory . . . . . . . . . . . . . . . . . . . . . . . . 828A.5 Dynamicists vision of turbulence . . . . . . . . . . . . . . . . . 833A.6 Gruppenpest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836A.7 Death of the Old Quantum Theory . . . . . . . . . . . . . . . . . 837commentary 840 references 841

B Go straight 849B.1 Rectification of flows . . . . . . . . . . . . . . . . . . . . . . . . 849B.2 Collinear helium . . . . . . . . . . . . . . . . . . . . . . . . . . 851B.3 Rectification of maps . . . . . . . . . . . . . . . . . . . . . . . . 855B.4 Rectification of a periodic orbit . . . . . . . . . . . . . . . . . . . 856resume 858 commentary 858 exercises 859 references 859

C Linear stability 861C.1 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 861C.2 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . 863C.3 Eigenspectra: what to make out of them? . . . . . . . . . . . . . . 872C.4 Stability of Hamiltonian flows . . . . . . . . . . . . . . . . . . . 872C.5 Monodromy matrix for Hamiltonian flows . . . . . . . . . . . . . 874exercises 877

D Discrete symmetries of dynamics 878D.1 Preliminaries and definitions . . . . . . . . . . . . . . . . . . . . 878D.2 Invariants and reducibility . . . . . . . . . . . . . . . . . . . . . 884D.3 Lattice derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 887D.4 Periodic lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . 891D.5 Discrete Fourier transforms . . . . . . . . . . . . . . . . . . . . . 892D.6 C4v factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . 896D.7 C2v factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . 900D.8 Henon map symmetries . . . . . . . . . . . . . . . . . . . . . . . 902commentary 903 exercises 903 references 905

E Finding cycles 908E.1 Newton-Raphson method . . . . . . . . . . . . . . . . . . . . . . 908E.2 Hybrid Newton-Raphson / relaxation method . . . . . . . . . . . 909

F Symbolic dynamics techniques 912F.1 Topological zeta functions for infinite subshifts . . . . . . . . . . 912

CONTENTS xii

F.2 Prime factorization for dynamical itineraries . . . . . . . . . . . . 920

G Counting itineraries 924G.1 Counting curvatures . . . . . . . . . . . . . . . . . . . . . . . . . 924exercises 925

H Implementing evolution 926H.1 Koopmania . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926H.2 Implementing evolution . . . . . . . . . . . . . . . . . . . . . . . 928commentary 931 exercises 931 references 932

I Transport of vector fields 934I.1 Evolution operator for Lyapunov exponents . . . . . . . . . . . . 934I.2 Advection of vector fields by chaotic flows . . . . . . . . . . . . . 939commentary 943 exercises 943 references 943

J Convergence of spectral determinants 945J.1 Curvature expansions: geometric picture . . . . . . . . . . . . . . 945J.2 On importance of pruning . . . . . . . . . . . . . . . . . . . . . . 948J.3 Ma-the-matical caveats . . . . . . . . . . . . . . . . . . . . . . . 949J.4 Estimate of the nth cumulant . . . . . . . . . . . . . . . . . . . . 950J.5 Dirichlet series . . . . . . . . . . . . . . . . . . . . . . . . . . . 952commentary 953

K Infinite dimensional operators 954K.1 Matrix-valued functions . . . . . . . . . . . . . . . . . . . . . . . 954K.2 Operator norms . . . . . . . . . . . . . . . . . . . . . . . . . . . 956K.3 Trace class and Hilbert-Schmidt class . . . . . . . . . . . . . . . 957K.4 Determinants of trace class operators . . . . . . . . . . . . . . . . 959K.5 Von Koch matrices . . . . . . . . . . . . . . . . . . . . . . . . . 962K.6 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964exercises 966 references 966

L Thermodynamic formalism 968L.1 Renyi entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . 968L.2 Fractal dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 973resume 977 commentary 977 exercises 978 references 978

M Statistical mechanics recycled 980M.1 The thermodynamic limit . . . . . . . . . . . . . . . . . . . . . . 980M.2 Ising models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983M.3 Fisher droplet model . . . . . . . . . . . . . . . . . . . . . . . . 986M.4 Scaling functions . . . . . . . . . . . . . . . . . . . . . . . . . . 991M.5 Geometrization . . . . . . . . . . . . . . . . . . . . . . . . . . . 994resume 1001 commentary 1002 exercises 1002 references 1003

N Noise/quantum corrections 1005N.1 Periodic orbits as integrable systems . . . . . . . . . . . . . . . . 1005N.2 The Birkho normal form . . . . . . . . . . . . . . . . . . . . . 1009N.3 Bohr-Sommerfeld quantization of periodic orbits . . . . . . . . . 1010

CONTENTS xiii

N.4 Quantum calculation of corrections . . . . . . . . . . . . . . . . 1012references 1018

O Projects 1021O.1 Deterministic diusion, zig-zag map . . . . . . . . . . . . . . . . 1023references 1028O.2 Deterministic diusion, sawtooth map . . . . . . . . . . . . . . . 1029

CONTENTS xiv

ContributorsNo man but a blockhead ever wrote except for money

Samuel Johnson

This book is a result of collaborative labors of many people over a span of severaldecades. Coauthors of a chapter or a section are indicated in the byline to thechapter/section title. If you are referring to a specific coauthored section ratherthan the entire book, cite it as (for example):

C. Chandre, F.K. Diakonos and P. Schmelcher, section Discrete cyclist re-laxation method, in P. Cvitanovic, R. Artuso, R. Mainieri, G. Tanner andG. Vattay, Chaos: Classical and Quantum (Niels Bohr Institute, Copen-hagen 2010); ChaosBook.org/version13.

Do not cite chapters by their numbers, as those change from version to version.Chapters without a byline are written by Predrag Cvitanovic. Friends whose con-tributions and ideas were invaluable to us but have not contributed written text tothis book, are credited in the acknowledgments.

Roberto Artuso19 Transporting densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35821.2 A trace formula for flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40327.3 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51629 Intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55124 Deterministic diusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

Ronnie Mainieri2 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 The Poincare section of a flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 Local stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84B.1 Understanding flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85014.1 Temporal ordering: itineraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248Appendix A: A brief history of chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821

Gabor VattayGregor Tanner

29 Intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551Appendix C.5: Jacobians of Hamiltonian flows . . . . . . . . . . . . . . . . . . . . 874

Arindam BasuRossler flow figures, tables, cycles in chapters 14, 16 and exercise 7.1

Ofer Biham33.1 Cyclists relaxation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649

Daniel Borrero Oct 23 2008, soluCycles.tex

Solution 16.1N. Burak Budanur

CONTENTS xv

0.113.5 First Fourier mode slice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22913.6 Stability within a slice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229Solution 12.5Solution 12.6

Cristel Chandre33.1 Cyclists relaxation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64933.2 Discrete cyclists relaxation methods . . . . . . . . . . . . . . . . . . . . . . . . . 654

Freddy Christiansen

7.1 One-dimensional mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13416.2 Multipoint shooting method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .307

Per Dahlqvist

29 Intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55133.3 Orbit length extremization method for billiards . . . . . . . . . . . . . . . 658

Carl P. Dettmann23.7 Stability ordering of cycle expansions . . . . . . . . . . . . . . . . . . . . . . . .443

Fotis K. Diakonos33.2 Discrete cyclists relaxation methods . . . . . . . . . . . . . . . . . . . . . . . . . 654

G. Bard ErmentroutExercise 5.1

Mitchell J. FeigenbaumAppendix C.4: Symplectic invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872

Sarah Flynn

solutions 3.5 and 3.6Matjaz Gomilsek

17.3 Transition graphs: stroll from link to link . . . . . . . . . . . . . . . . . . . . .323

Jonathan HalcrowExample 3.4: Sections of Lorenz flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Example 4.6: Stability of Lorenz flow equilibria . . . . . . . . . . . . . . . . . . . 100Example 4.7: Lorenz flow: Global portrait . . . . . . . . . . . . . . . . . . . . . . . . 102Example 11.5: Desymmetrization of Lorenz flow . . . . . . . . . . . . . . . . . . 194Example 14.5: Lorenz flow: a 1-dimensional return map . . . . . . . . . . . 266Exercises 11.4 and figure 2.5

Kai T. Hansen14.3 Unimodal map symbolic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 25418.5 Topological zeta function for an infinite partition . . . . . . . . . . . . . .34114.5 Kneading theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257figures throughout the text

CONTENTS xvi

Rainer Klages

Figure 24.5

Yueheng Lan

Solutions 1.1, 2.2, 2.3, 2.4, 2.5, 11.1, 15.7, 14.6, 19.1, 19.2, 19.3, 19.5,19.7, 19.10, 6.3 and figures 1.9, 11.1, 11.5 14.5,

Bo LiSolutions 35.2, 35.1, 36.1

Norman LebovitzExample 15.2 A simple stable/unstable manifolds pair . . . . . . . . . . . . . 294

Joachim Mathiesen6.2 Lyapunov exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Rossler flow figures, tables, cycles in sect. 6.2 and exercise 7.1

Yamato MatsuokaFigure 15.4

Radford Mitchell, Jr.Example 3.5

Rytis Paskauskas

4.6 Stability of Poincare return maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.5 Stability of Poincare map cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Exercises 2.8, 3.1, 4.4 and solution 4.1

Adam Prugel-Bennet

Solutions 1.2, 2.10, 9.1, 20.2, 23.2 28.3, 33.1.Lamberto Rondoni

19 Transporting densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35816.1.1 Cycles from long time series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30527.2.1 Unstable periodic orbits are dense . . . . . . . . . . . . . . . . . . . . . . . . . 515Table 18.2

Juri RolfSolution 28.3

Per E. Rosenqvist

exercises, figures throughout the text

Hans Henrik Rugh

28 Why does it work? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

Luis Saldanasolution 11.2

Peter Schmelcher33.2 Discrete cyclists relaxation methods . . . . . . . . . . . . . . . . . . . . . . . . . 654

CONTENTS xvii

Evangelos Siminos

Example 3.4: Sections of Lorenz flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Example 4.6: Stability of Lorenz flow equilibria . . . . . . . . . . . . . . . . . . . 100Example 4.7: Lorenz flow: Global portrait . . . . . . . . . . . . . . . . . . . . . . . . 102Example 11.5: Desymmetrization of Lorenz flow . . . . . . . . . . . . . . . . . . 194Example 14.5: Lorenz flow: a 1-dimensional return map . . . . . . . . . . . 266Exercise 11.4Solution 13.1

Gabor SimonRossler flow figures, tables, cycles in chapters 2, 16 and exercise 7.1

Edward A. Spiegel

2 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3719 Transporting densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

Luz V. Vela-Arevalo8.1 Hamiltonian flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145Exercises 8.1, 8.3, 8.5

Lei Zhang

Solutions 1.1, 2.1

CONTENTS xviii

Acknowledgments

I feel I never want to write another book. Whats the good!I can eke living on stories and little articles, that dont costa tithe of the output a book costs. Why write novels anymore!

D.H. Lawrence

This book owes its existence to the Niels Bohr Institutes and Norditas hos-pitable and nurturing environment, and the private, national and cross-nationalfoundations that have supported the collaborators research over a span of severaldecades. P.C. thanks M.J. Feigenbaum of Rockefeller University; D. Ruelle ofI.H.E.S., Bures-sur-Yvette; I. Procaccia of Minerva Center for Nonlinear Physicsof Complex Systems, Weizmann Institute of Science; P.H. Damgaard of the NielsBohr International Academy; G. Mazenko of U. of Chicago James Franck Insti-tute and Argonne National Laboratory; T. Geisel of Max-Planck-Institut fur Dy-namik und Selbstorganisation, Gottingen; I. Andric of Rudjer Boskovic Institute;P. Hemmer of University of Trondheim; The Max-Planck Institut fur Mathematik,Bonn; J. Lowenstein of New York University; Edificio Celi, Milano; Fundacaode Faca, Porto Seguro; and Dr. Dj. Cvitanovic, Kostrena, for the hospitality dur-ing various stages of this work, and the Carlsberg Foundation, Glen P. Robinson,Humboldt Foundation and National Science Fundation grant DMS-0807574 forpartial support.

The authors gratefully acknowledge collaborations and/or stimulating discus-sions with E. Aurell, M. Avila, V. Baladi, D. Barkley, B. Brenner, G. Byrne,A. de Carvalho, D.J. Driebe, B. Eckhardt, M.J. Feigenbaum, J. Frjland, S. Froehlich,P. Gaspar, P. Gaspard, J. Guckenheimer, G.H. Gunaratne, P. Grassberger, H. Gutowitz,M. Gutzwiller, K.T. Hansen, P.J. Holmes, T. Janssen, R. Klages, T. Kreilos, Y. Lan,B. Lauritzen, C. Marcotte, J. Milnor, M. Nordahl, I. Procaccia, J.M. Robbins,P.E. Rosenqvist, D. Ruelle, G. Russberg, B. Sandstede, A. Shapere, M. Sieber,D. Sullivan, N. Sndergaard, T. Tel, C. Tresser, R. Wilczak, and D. Wintgen.

We thank Dorte Glass, Tzatzilha Torres Guadarrama and Raenell Soller fortyping parts of the manuscript; D. Borrero, P. Duren, B. Lautrup, J.F Gibson,M. Gomilsek and D. Viswanath for comments and corrections to the prelimi-nary versions of this text; M.A. Porter for patiently and critically reading themanuscript, and then lengthening by the 2013 definite articles hitherto missing;M.V. Berry for the quotation on page 821; H. Fogedby for the quotation onpage 534; J. Greensite for the quotation on page 7; S. Ortega Arango for the quo-tation on page 16; Ya.B. Pesin for the remarks quoted on page 841; M.A. Porterfor the quotations on pages 8.1, 20, 16, 1.6 and A.4; E.A. Spiegel for quotationon page 3; and E. Valesco for the quotation on page 25.

F. Haakes heartfelt lament on page 403 was uttered at the end of the firstconference presentation of cycle expansions, in 1988. G.P. Morriss advice tostudents as how to read the introduction to this book, page6, was oered duringa 2002 graduate course in Dresden. J. Bellissards advice to students concerningunpleasant operators and things nonlinear, pages4.3 and 20.3.1, was shared in his2013 Classical Mechanics II lectures on manifolds. K. Huangs C.N. Yang in-terview quoted on page 365 is available on ChaosBook.org/extras. T.D. Leeremarks on as to who is to blame, page 37 and page 305, as well as M. Shubs

CONTENTS xix

helpful technical remark on page 546 came during the Rockefeller University De-cember 2004 Feigenbaum Fest. Quotes on pages 37, 144, and 362 are takenfrom a book review by J. Guckenheimer [1].

Who is the 3-legged dog reappearing throughout the book? Long ago, whenwe were innocent and knew not Borel measurable to sets, P. Cvitanovic askedV. Baladi a question about dynamical zeta functions, who then asked J.-P. Eck-mann, who then asked D. Ruelle. The answer was transmitted back: The mastersays: It is holomorphic in a strip. Hence His Masters Voice logo, and the 3-legged dog is us, still eager to fetch the bone. The answer has made it to the book,though not precisely in His Masters voice. As a matter of fact, the answer is thebook. We are still chewing on it.

What about the two beers? During his PhD studies, R. Artuso found thesmrrebrd at the Niels Bohr Institute indigestible, so he digested H.M.V.s wis-dom on a strict diet of two Carlsbergs and two pieces of danish pastry for lunchevery day, as depicted on the cover. Frequent trips back to Milano family kept himalivehe never got desperate enough to try the Danish smrrebrd. And the cyclewheel? Well, this is no book for pedestrians.

And last but not least: profound thanks to all the unsung heroes students andcolleagues, too numerous to list here who have supported this project over manyyears in many ways, by surviving pilot courses based on this book, by providinginvaluable insights, by teaching us, by inspiring us.

Part I

Geometry of chaos

1

2We start out with a recapitulation of the basic notions of dynamics. Our aim isnarrow; we keep the exposition focused on prerequisites to the applications tobe developed in this text. We assume that the reader is familiar with dynamicson the level of the introductory texts mentioned in remark 1.1, and concentrate here ondeveloping intuition about what a dynamical system can do. It will be a broad strokedescription, since describing all possible behaviors of dynamical systems is beyondhuman ken. While for a novice there is no shortcut through this lengthy detour, asophisticated traveler might bravely skip this well-trodden territory and embark upon thejourney at chapter 18.

The fate has handed you a flow. What are you to do about it?

1. Define your dynamical system (M, f ): the space M of its possible states, and thelaw f t of their evolution in time.

2. Pin it down locallyis there anything about it that is stationary? Try to determine itsequilibria / fixed points (chapter 2).

3. Cut across it, represent as a Poincare map from a section to a section (chapter3).4. Explore the neighborhood by linearizing the flow; check the linear stability of its

equilibria / fixed points, their stability eigen-directions (chapters4 and 5).5. Does your system have a symmetry? If so, you must use it (chapters10 to 12). Slice

& dice it (chapter 13).6. Go global: train by partitioning the state space of 1-dimensional maps. Label the

regions by symbolic dynamics (chapter 14).7. Now venture global distances across the system by continuing local tangent space

into stable / unstable manifolds. Their intersections partition the state space in adynamically invariant way (chapter 15).

8. Guided by this topological partition, compute a set of periodic orbits up to a giventopological length (chapter 7 and chapter 16).

Along the way you might want to learn about Lyapunov exponents (chapter6), classicalmechanics (chapter 8), and billiards (chapter 9).

ackn.tex 12decd2010ChaosBook.org version15.6, Mar 15 2015

Chapter 1

Overture

If I have seen less far than other men it is because I havestood behind giants.

Edoardo Specchio

Rereading classic theoretical physics textbooks leaves a sense that there areholes large enough to steam a Eurostar train through them. Here we learnabout harmonic oscillators and Keplerian ellipses - but where is the chap-ter on chaotic oscillators, the tumbling Hyperion? We have just quantized hydro-gen, where is the chapter on the classical 3-body problem and its implications forquantization of helium? We have learned that an instanton is a solution of field-theoretic equations of motion, but shouldnt a strongly nonlinear field theory haveturbulent solutions? How are we to think about systems where things fall apart;the center cannot hold; every trajectory is unstable?

This chapter oers a quick survey of the main topics covered in the book.Throughout the book

indicates that the section is on a pedestrian level - you are expected toknow/learn this material

indicates that the section is on a somewhat advanced, cyclist level

indicates that the section requires a hearty stomach and is probably bestskipped on first reading

fast track points you where to skip to

tells you where to go for more depth on a particular topic

link to a related video

[exercise 1.2] on margin links to an exercise that might clarify a point in the text

3

CHAPTER 1. OVERTURE 4

indicates that a figure is still missingyou are urged to fetch it

We start out by making promiseswe will right wrongs, no longer shall you suerthe slings and arrows of outrageous Science of Perplexity. We relegate a historicaloverview of the development of chaotic dynamics to appendixA, and head straightto the starting line: A pinball game is used to motivate and illustrate most of theconcepts to be developed in ChaosBook.

This is a textbook, not a research monograph, and you should be able to followthe thread of the argument without constant excursions to sources. Hence there areno literature references in the text proper, all learned remarks and bibliographicalpointers are relegated to the Commentary section at the end of each chapter.

1.1 Why ChaosBook?

It seems sometimes that through a preoccupation with sci-ence, we acquire a firmer hold over the vicissitudes of lifeand meet them with greater calm, but in reality we havedone no more than to find a way to escape from our sor-rows.

Hermann Minkowski in a letter to David Hilbert

The problem has been with us since Newtons first frustrating (and unsuccessful)crack at the 3-body problem, lunar dynamics. Nature is rich in systems governedby simple deterministic laws whose asymptotic dynamics are complex beyondbelief, systems which are locally unstable (almost) everywhere but globally recur-rent. How do we describe their long term dynamics?

The answer turns out to be that we have to evaluate a determinant, take alogarithm. It would hardly merit a learned treatise, were it not for the fact that thisdeterminant that we are to compute is fashioned out of infinitely many infinitelysmall pieces. The feel is of statistical mechanics, and that is how the problemwas solved; in the 1960s the pieces were counted, and in the 1970s they wereweighted and assembled in a fashion that in beauty and in depth ranks along withthermodynamics, partition functions and path integrals amongst the crown jewelsof theoretical physics.

This book is not a book about periodic orbits. The red thread throughout thetext is the duality between the local, topological, short-time dynamically invariantcompact sets (equilibria, periodic orbits, partially hyperbolic invariant tori) andthe global long-time evolution of densities of trajectories. Chaotic dynamics isgenerated by the interplay of locally unstable motions, and the interweaving oftheir global stable and unstable manifolds. These features are robust and acces-sible in systems as noisy as slices of rat brains. Poincare, the first to understanddeterministic chaos, already said as much (modulo rat brains). Once this topology

intro - 9apr2009 ChaosBook.org version15.6, Mar 15 2015


is understood, a powerful theory yields the observable consequences of chaoticdynamics, such as atomic spectra, transport coecients, gas pressures.

That is what we will focus on in ChaosBook. The book is a self-containedgraduate textbook on classical and quantum chaos. Your professor does not knowthis material, so you are on your own. We will teach you how to evaluate a deter-minant, take a logarithmstu like that. Ideally, this should take 100 pages or so.Well, we failso far we have not found a way to traverse this material in less thana semester, or 200-300 page subset of this text. Nothing to be done.

1.2 Chaos ahead

Things fall apart; the centre cannot hold.W.B. Yeats, The Second Coming

The study of chaotic dynamics is no recent fashion. It did not start with thewidespread use of the personal computer. Chaotic systems have been studied forover 200 years. During this time many have contributed, and the field followed nosingle line of development; rather one sees many interwoven strands of progress.

In retrospect many triumphs of both classical and quantum physics were astroke of luck: a few integrable problems, such as the harmonic oscillator andthe Kepler problem, though non-generic, have gotten us very far. The successhas lulled us into a habit of expecting simple solutions to simple equationsanexpectation tempered by our recently acquired ability to numerically scan the statespace of non-integrable dynamical systems. The initial impression might be thatall of our analytic tools have failed us, and that the chaotic systems are amenableonly to numerical and statistical investigations. Nevertheless, a beautiful theoryof deterministic chaos, of predictive quality comparable to that of the traditionalperturbation expansions for nearly integrable systems, already exists.

In the traditional approach the integrable motions are used as zeroth-order ap-proximations to physical systems, and weak nonlinearities are then accounted forperturbatively. For strongly nonlinear, non-integrable systems such expansionsfail completely; at asymptotic times the dynamics exhibits amazingly rich struc-ture which is not at all apparent in the integrable approximations. However, hiddenin this apparent chaos is a rigid skeleton, a self-similar tree of cycles (periodic or-bits) of increasing lengths. The insight of the modern dynamical systems theoryis that the zeroth-order approximations to the harshly chaotic dynamics should bevery dierent from those for the nearly integrable systems: a good starting ap-proximation here is the stretching and folding of bakers dough, rather than theperiodic motion of a harmonic oscillator.

So, what is chaos, and what is to be done about it? To get some feeling for howand why unstable cycles come about, we start by playing a game of pinball. Theremainder of the chapter is a quick tour through the material covered in Chaos-Book. Do not worry if you do not understand every detail at the first readingthe



Figure 1.1: A physicists bare bones game of pinball.

intention is to give you a feeling for the main themes of the book. Details willbe filled out later. If you want to get a particular point clarified right now, [section

section 1.41.4] on the margin points at the appropriate section.

1.3 The future as in a mirror

All you need to know about chaos is contained in the intro-duction of [ChaosBook]. However, in order to understandthe introduction you will first have to read the rest of thebook.

Gary Morriss

That deterministic dynamics leads to chaos is no surprise to anyone who has triedpool, billiards or snookerthe game is about beating chaosso we start our storyabout what chaos is, and what to do about it, with a game of pinball. This mightseem a trifle, but the game of pinball is to chaotic dynamics what a pendulum isto integrable systems: thinking clearly about what chaos in a game of pinballis will help us tackle more dicult problems, such as computing the diusionconstant of a deterministic gas, the drag coecient of a turbulent boundary layer,or the helium spectrum.

We all have an intuitive feeling for what a ball does as it bounces among thepinball machines disks, and only high-school level Euclidean geometry is neededto describe its trajectory. A physicists pinball game is the game of pinball strip-ped to its bare essentials: three equidistantly placed reflecting disks in a plane,figure 1.1. A physicists pinball is free, frictionless, point-like, spin-less, perfectlyelastic, and noiseless. Point-like pinballs are shot at the disks from random startingpositions and angles; they spend some time bouncing between the disks and thenescape.

At the beginning of the 18th century Baron Gottfried Wilhelm Leibniz wasconfident that given the initial conditions one knew everything a deterministicsystem would do far into the future. He wrote [2], anticipating by a century anda half the oft-quoted Laplaces Given for one instant an intelligence which couldcomprehend all the forces by which nature is animated...:

That everything is brought forth through an established destiny is just



Figure 1.2: Sensitivity to initial conditions: two pin-balls that start out very close to each other separate ex-ponentially with time.

1

2

3

23132321

2313

as certain as that three times three is nine. [. . . ] If, for example, one spheremeets another sphere in free space and if their sizes and their paths anddirections before collision are known, we can then foretell and calculatehow they will rebound and what course they will take after the impact. Verysimple laws are followed which also apply, no matter how many spheresare taken or whether objects are taken other than spheres. From this onesees then that everything proceeds mathematicallythat is, infalliblyin thewhole wide world, so that if someone could have a sucient insight intothe inner parts of things, and in addition had remembrance and intelligenceenough to consider all the circumstances and to take them into account, hewould be a prophet and would see the future in the present as in a mirror.

Leibniz chose to illustrate his faith in determinism precisely with the type of phys-ical system that we shall use here as a paradigm of chaos. His claim is wrong in adeep and subtle way: a state of a physical system can never be specified to infiniteprecision, and by this we do not mean that eventually the Heisenberg uncertaintyprinciple kicks in. In the classical, deterministic dynamics there is no way to takeall the circumstances into account, and a single trajectory cannot be tracked, onlya ball of nearby initial points makes physical sense.

1.3.1 What is chaos?

I accept chaos. I am not sure that it accepts me.Bob Dylan, Bringing It All Back Home

A deterministic system is a system whose present state is in principle fully deter-mined by its initial conditions.

In contrast, radioactive decay, Brownian motion and heat flow are examplesof stochastic systems, for which the initial conditions determine the future onlypartially, due to noise, or other external circumstances beyond our control: thepresent state reflects the past initial conditions plus the particular realization ofthe noise encountered along the way.

A deterministic system with suciently complicated dynamics can appear tous to be stochastic; disentangling the deterministic from the stochastic is the main



Figure 1.3: Unstable trajectories separate with time. x(0)

x(t)

x(t)x(0)

challenge in many real-life settings, from stock markets to palpitations of chickenhearts. So, what is chaos?

In a game of pinball, any two trajectories that start out very close to each otherseparate exponentially with time, and in a finite (and in practice, a very small)number of bounces their separation x(t) attains the magnitude of L, the charac-teristic linear extent of the whole system, figure 1.2. This property of sensitivityto initial conditions can be quantified as

|x(t)| et |x(0)|

where , the mean rate of separation of trajectories of the system, is called theLyapunov exponent. For any finite accuracy x = |x(0)| of the initial data, the

chapter 6dynamics is predictable only up to a finite Lyapunov time

TLyap 1 ln |x/L| , (1.1)

despite the deterministic and, for Baron Leibniz, infallible simple laws that rulethe pinball motion.

A positive Lyapunov exponent does not in itself lead to chaos. One could tryto play 1- or 2-disk pinball game, but it would not be much of a game; trajecto-ries would only separate, never to meet again. What is also needed is mixing, thecoming together again and again of trajectories. While locally the nearby trajec-tories separate, the interesting dynamics is confined to a globally finite region ofthe state space and thus the separated trajectories are necessarily folded back andcan re-approach each other arbitrarily closely, infinitely many times. For the caseat hand there are 2n topologically distinct n bounce trajectories that originate froma given disk. More generally, the number of distinct trajectories with n bouncescan be quantified as

section 18.1

N(n) ehn

where h, the growth rate of the number of topologically distinct trajectories, iscalled the topological entropy (h = ln 2 in the case at hand).

The appellation chaos is a confusing misnomer, as in deterministic dynam-ics there is no chaos in the everyday sense of the word; everything proceedsmathematicallythat is, as Baron Leibniz would have it, infallibly. When a physi-cist says that a certain system exhibits chaos, he means that the system obeys



Figure 1.4: Dynamics of a chaotic dynamical sys-tem is (a) everywhere locally unstable (positiveLyapunov exponent) and (b) globally mixing (pos-itive entropy). (A. Johansen)

(a) (b)

deterministic laws of evolution, but that the outcome is highly sensitive to smalluncertainties in the specification of the initial state. The word chaos has in thiscontext taken on a narrow technical meaning. If a deterministic system is locallyunstable (positive Lyapunov exponent) and globally mixing (positive entropy)figure 1.4it is said to be chaotic.

While mathematically correct, the definition of chaos as positive Lyapunov+ positive entropy is useless in practice, as a measurement of these quantities isintrinsically asymptotic and beyond reach for systems observed in nature. Morepowerful is Poincares vision of chaos as the interplay of local instability (unsta-ble periodic orbits) and global mixing (intertwining of their stable and unstablemanifolds). In a chaotic system any open ball of initial conditions, no matter howsmall, will in a finite time overlap with any other finite region and in this sensespread over the extent of the entire asymptotically accessible state space. Oncethis is grasped, the focus of theory shifts from attempting to predict individualtrajectories (which is impossible) to a description of the geometry of the spaceof possible outcomes, and evaluation of averages over this space. How this isaccomplished is what ChaosBook is about.

A definition of turbulence is even harder to come by. Can you recognizeturbulence when you see it? The word comes from tourbillon, French for vor-tex, and intuitively it refers to irregular behavior of spatially extended systemdescribed by deterministic equations of motionsay, a bucket of sloshing waterdescribed by the Navier-Stokes equations. But in practice the word turbulencetends to refer to messy dynamics which we understand poorly. As soon as aphenomenon is understood better, it is reclaimed and renamed: a route to chaos,spatiotemporal chaos, and so on.

In ChaosBook we shall develop a theory of chaotic dynamics for low dimens-ional attractors visualized as a succession of nearly periodic but unstable motions.In the same spirit, we shall think of turbulence in spatially extended systems interms of recurrent spatiotemporal patterns. Pictorially, dynamics drives a givenspatially extended system (clouds, say) through a repertoire of unstable patterns;as we watch a turbulent system evolve, every so often we catch a glimpse of afamiliar pattern:

= other swirls =



For any finite spatial resolution, a deterministic flow follows approximately for afinite time an unstable pattern belonging to a finite alphabet of admissible patterns,and the long term dynamics can be thought of as a walk through the space of suchpatterns. In ChaosBook we recast this image into mathematics.

1.3.2 When does chaos matter?

In dismissing Pollocks fractals because of their limitedmagnification range, Jones-Smith and Mathur would alsodismiss half the published investigations of physical frac-tals.

Richard P. Taylor [4, 5]

When should we be mindful of chaos? The solar system is chaotic, yet wehave no trouble keeping track of the annual motions of planets. The rule of thumbis this; if the Lyapunov time (1.1)the time by which a state space region initiallycomparable in size to the observational accuracy extends across the entire acces-sible state spaceis significantly shorter than the observational time, you need tomaster the theory that will be developed here. That is why the main successes ofthe theory are in statistical mechanics, quantum mechanics, and questions of longterm stability in celestial mechanics.

In science popularizations too much has been made of the impact of chaostheory, so a number of caveats are already needed at this point.

At present the theory that will be developed here is in practice applicable onlyto systems of a low intrinsic dimension the minimum number of coordinates nec-essary to capture its essential dynamics. If the system is very turbulent (a descrip-tion of its long time dynamics requires a space of high intrinsic dimension) we areout of luck. Hence insights that the theory oers in elucidating problems of fullydeveloped turbulence, quantum field theory of strong interactions and early cos-mology have been modest at best. Even that is a caveat with qualifications. Thereare applicationssuch as spatially extended (non-equilibrium) systems, plumbersturbulent pipes, etc.,where the few important degrees of freedom can be isolatedand studied profitably by methods to be described here.

Thus far the theory has had limited practical success when applied to the verynoisy systems so important in the life sciences and in economics. Even thoughwe are often interested in phenomena taking place on time scales much longerthan the intrinsic time scale (neuronal inter-burst intervals, cardiac pulses, etc.),disentangling chaotic motions from the environmental noise has been very hard.

In 1980s something happened that might be without parallel; this is an areaof science where the advent of cheap computation had actually subtracted fromour collective understanding. The computer pictures and numerical plots of frac-tal science of the 1980s have overshadowed the deep insights of the 1970s, andthese pictures have since migrated into textbooks. By a regrettable oversight,



Figure 1.5: Katherine Jones-Smith, Untitled 5, thedrawing used by K. Jones-Smith and R.P. Taylor to testthe fractal analysis of Pollocks drip paintings [6].

ChaosBook has none, so Untitled 5 of figure 1.5 will have to do as the illustra-tion of the power of fractal analysis. Fractal science posits that certain quantities

remark 1.7(Lyapunov exponents, generalized dimensions, . . . ) can be estimated on a com-puter. While some of the numbers so obtained are indeed mathematically sensiblecharacterizations of fractals, they are in no sense observable and measurable onthe length-scales and time-scales dominated by chaotic dynamics.

Even though the experimental evidence for the fractal geometry of nature iscircumstantial [7], in studies of probabilistically assembled fractal aggregates weknow of nothing better than contemplating such quantities. In deterministic sys-tems we can do much better.

1.4 A game of pinball

Formulas hamper the understanding.S. Smale

We are now going to get down to the brass tacks. Time to fasten your seat beltsand turn o all electronic devices. But first, a disclaimer: If you understand therest of this chapter on the first reading, you either do not need this book, or you aredelusional. If you do not understand it, it is not because the people who figuredall this out first are smarter than you: the most you can hope for at this stage is toget a flavor of what lies ahead. If a statement in this chapter mystifies/intrigues,fast forward to a section indicated by [section ...] on the margin, read only theparts that you feel you need. Of course, we think that you need to learn ALL of it,or otherwise we would not have included it in ChaosBook in the first place.

Confronted with a potentially chaotic dynamical system, our analysis pro-ceeds in three stages; I. diagnose, II. count, III. measure. First, we determine



Figure 1.6: Binary labeling of the 3-disk pinball tra-jectories; a bounce in which the trajectory returns tothe preceding disk is labeled 0, and a bounce whichresults in continuation to the third disk is labeled 1.

the intrinsic dimension of the systemthe minimum number of coordinates nec-essary to capture its essential dynamics. If the system is very turbulent we are,at present, out of luck. We know only how to deal with the transitional regimebetween regular motions and chaotic dynamics in a few dimensions. That is stillsomething; even an infinite-dimensional system such as a burning flame front canturn out to have a very few chaotic degrees of freedom. In this regime the chaoticdynamics is restricted to a space of low dimension, the number of relevant param-eters is small, and we can proceed to step II; we count and classify all possible

chapter 14chapter 18topologically distinct trajectories of the system into a hierarchy whose successive

layers require increased precision and patience on the part of the observer. Thiswe shall do in sect. 1.4.2. If successful, we can proceed with step III: investigatethe weights of the dierent pieces of the system.

We commence our analysis of the pinball game with steps I, II: diagnose,count. We shall return to step IIImeasurein sect. 1.5. The three sections that

chapter 23follow are highly technical, they go into the guts of what the book is about. Iftoday is not your thinking day, skip them, jump straight to sect.1.7.

1.4.1 Symbolic dynamics

With the game of pinball we are in luckit is a low dimensional system, freemotion in a plane. The motion of a point particle is such that after a collisionwith one disk it either continues to another disk or it escapes. If we label thethree disks by 1, 2 and 3, we can associate every trajectory with an itinerary, asequence of labels indicating the order in which the disks are visited; for example,the two trajectories in figure 1.2 have itineraries 2313 , 23132321 respectively.

exercise 1.1section 2.1Such labeling goes by the name symbolic dynamics. As the particle cannot collide

two times in succession with the same disk, any two consecutive symbols mustdier. This is an example of pruning, a rule that forbids certain subsequencesof symbols. Deriving pruning rules is in general a dicult problem, but with thegame of pinball we are luckyfor well-separated disks there are no further pruningrules.

chapter 15

The choice of symbols is in no sense unique. For example, as at each bouncewe can either proceed to the next disk or return to the previous disk, the above3-letter alphabet can be replaced by a binary {0, 1} alphabet, figure1.6. A cleverchoice of an alphabet will incorporate important features of the dynamics, such asits symmetries.

section 14.6

Suppose you wanted to play a good game of pinball, that is, get the pinballto bounce as many times as you possibly canwhat would be a winning strategy?The simplest thing would be to try to aim the pinball so it bounces many times



Figure 1.7: The 3-disk pinball cycles 12323 and121212313.

Figure 1.8: (a) A trajectory starting out from disk1 can either hit another disk or escape. (b) Hittingtwo disks in a sequence requires a much sharper aim,with initial conditions that hit further consecutive disksnested within each other, as in Fig. 1.9.

between a pair of disksif you managed to shoot it so it starts out in the periodicorbit bouncing along the line connecting two disk centers, it would stay there for-ever. Your game would be just as good if you managed to get it to keep bouncingbetween the three disks forever, or place it on any periodic orbit. The only rubis that any such orbit is unstable, so you have to aim very accurately in order tostay close to it for a while. So it is pretty clear that if one is interested in playingwell, unstable periodic orbits are importantthey form the skeleton onto which alltrajectories trapped for long times cling.

1.4.2 Partitioning with periodic orbits

A trajectory is periodic if it returns to its starting position and momentum. Weshall sometimes refer to the set of periodic points that belong to a given periodicorbit as a cycle.

Short periodic orbits are easily drawn and enumeratedan example is drawn infigure 1.7but it is rather hard to perceive the systematics of orbits from their con-figuration space shapes. In mechanics a trajectory is fully and uniquely specifiedby its position and momentum at a given instant, and no two distinct state spacetrajectories can intersect. Their projections onto arbitrary subspaces, however,can and do intersect, in rather unilluminating ways. In the pinball example theproblem is that we are looking at the projections of a 4-dimensional state spacetrajectories onto a 2-dimensional subspace, the configuration space. A clearerpicture of the dynamics is obtained by constructing a set of state space Poincaresections.

Suppose that the pinball has just bounced o disk 1. Depending on its positionand outgoing angle, it could proceed to either disk 2 or 3. Not much happens inbetween the bouncesthe ball just travels at constant velocity along a straight lineso we can reduce the 4-dimensional flow to a 2-dimensional map P that takes thecoordinates of the pinball from one disk edge to another disk edge. The trajectoryjust after the moment of impact is defined by sn, the arc-length position of thenth bounce along the billiard wall, and pn = p sin n the momentum component



Figure 1.9: The 3-disk game of pinball Poincaresection, trajectories emanating from the disk 1with x0 = (s0, p0) . (a) Strips of initial pointsM12,M13 which reach disks 2, 3 in one bounce, respec-tively. (b) Strips of initial pointsM121,M131 M132andM123 which reach disks 1, 2, 3 in two bounces,respectively. The Poincare sections for trajectoriesoriginating on the other two disks are obtained bythe appropriate relabeling of the strips. Disk ra-dius : center separation ratio a:R = 1:2.5. (Y.Lan)

(a)

sin

1

0

12.5

S0 2.5

1312

(b)

1

0

sin

1

2.50s

2.5

132

131123

121

parallel to the billiard wall at the point of impact, see figure1.9. Such section of aflow is called a Poincare section. In terms of Poincare sections, the dynamics is

example 15.9reduced to the set of six maps Psks j : (sn, pn) (sn+1, pn+1), with s {1, 2, 3},from the boundary of the disk j to the boundary of the next disk k.

chapter 9

Next, we mark in the Poincare section those initial conditions which do notescape in one bounce. There are two strips of survivors, as the trajectories orig-inating from one disk can hit either of the other two disks, or escape withoutfurther ado. We label the two strips M12, M13. Embedded within them thereare four stripsM121,M123,M131,M132 of initial conditions that survive for twobounces, and so forth, see figures 1.8 and 1.9. Provided that the disks are su-ciently separated, after n bounces the survivors are divided into 2n distinct strips:the Mith strip consists of all points with itinerary i = s1s2s3 . . . sn, s = {1, 2, 3}.The unstable cycles as a skeleton of chaos are almost visible here: each such patchcontains a periodic point s1s2s3 . . . sn with the basic block infinitely repeated. Pe-riodic points are skeletal in the sense that as we look further and further, the stripsshrink but the periodic points stay put forever.

We see now why it pays to utilize a symbolic dynamics; it provides a naviga-tion chart through chaotic state space. There exists a unique trajectory for everyadmissible infinite length itinerary, and a unique itinerary labels every trappedtrajectory. For example, the only trajectory labeled by 12 is the 2-cycle bouncingalong the line connecting the centers of disks 1 and 2; any other trajectory startingout as 12 . . . either eventually escapes or hits the 3rd disk.

1.4.3 Escape rateexample 20.4

What is a good physical quantity to compute for the game of pinball? Such a sys-tem, for which almost any trajectory eventually leaves a finite region (the pinballtable) never to return, is said to be open, or a repeller. The repeller escape rateis an eminently measurable quantity. An example of such a measurement wouldbe an unstable molecular or nuclear state which can be well approximated by aclassical potential with the possibility of escape in certain directions. In an ex-periment many projectiles are injected into a macroscopic black box enclosinga microscopic non-confining short-range potential, and their mean escape rate ismeasured, as in figure 1.1. The numerical experiment might consist of injecting



the pinball between the disks in some random direction and asking how manytimes the pinball bounces on the average before it escapes the region between thedisks.

exercise 1.2

For a theorist, a good game of pinball consists in predicting accurately theasymptotic lifetime (or the escape rate) of the pinball. We now show how periodicorbit theory accomplishes this for us. Each step will be so simple that you canfollow even at the cursory pace of this overview, and still the result is surprisinglyelegant.

Consider figure 1.9 again. In each bounce the initial conditions get thinnedout, yielding twice as many thin strips as at the previous bounce. The total areathat remains at a given time is the sum of the areas of the strips, so that the fractionof survivors after n bounces, or the survival probability is given by

1 =|M0||M| +

|M1||M| ,

2 =|M00||M| +

|M10||M| +

|M01||M| +

|M11||M| ,

n =1|M|

(n)i|Mi| , (1.2)

where i is a label of the ith strip, |M| is the initial area, and |Mi| is the area ofthe ith strip of survivors. i = 01, 10, 11, . . . is a label, not a binary number. Sinceat each bounce one routinely loses about the same fraction of trajectories, oneexpects the sum (1.2) to fall o exponentially with n and tend to the limit

chapter 27

n+1/ n = en e. (1.3)

The quantity is called the escape rate from the repeller.

1.5 Chaos for cyclists

Etant donnees des equations ... et une solution particulierequelconque de ces equations, on peut toujours trouver unesolution periodique (dont la periode peut, il est vrai, etretres longue), telle que la dierence entre les deux solu-tions soit aussi petite quon le veut, pendant un temps aussilong quon le veut. Dailleurs, ce qui nous rend ces solu-tions periodiques si precieuses, cest quelles sont, pouransi dire, la seule breche par ou` nous puissions esseyer depenetrer dans une place jusquici reputee inabordable.

H. Poincare, Les methodes nouvelles de lamechanique celeste

We shall now show that the escape rate can be extracted from a highly conver-gent exact expansion by reformulating the sum (1.2) in terms of unstable periodicorbits.



If, when asked what the 3-disk escape rate is for a disk of radius 1, center-center separation 6, velocity 1, you answer that the continuous time escape rateis roughly = 0.4103384077693464893384613078192 . . ., you do not need thisbook. If you have no clue, hang on.

1.5.1 How big is my neighborhood?

Of course, we can prove all these results directly fromEq. (20.25) by pedestrian mathematical manipulations,but that only makes it harder to appreciate their physicalsignificance.

Rick Salmon, Lectures on Geophysical Fluid Dy-namics, Oxford Univ. Press (1998)

Not only do the periodic points keep track of topological ordering of the strips,but, as we shall now show, they also determine their size. As a trajectory evolves,it carries along and distorts its infinitesimal neighborhood. Let

x(t) = f t(x0)

denote the trajectory of an initial point x0 = x(0). Expanding f t(x0 + x0) tolinear order, the evolution of the distance to a neighboring trajectory x(t) + x(t)is given by the Jacobian matrix J:

xi(t) =d

j=1Jt(x0)i jx0 j , Jt(x0)i j =

xi(t)x0 j

. (1.4)

A trajectory of a pinball moving on a flat surface is specified by two positioncoordinates and the direction of motion, so in this case d = 3. Evaluation ofa cycle Jacobian matrix is a long exercise - here we just state the result. The

section 9.2Jacobian matrix describes the deformation of an infinitesimal neighborhood ofx(t) along the flow; its eigenvectors and eigenvalues give the directions and thecorresponding rates of expansion or contraction, figure1.10. The trajectories thatstart out in an infinitesimal neighborhood separate along the unstable directions(those whose eigenvalues are greater than unity in magnitude), approach eachother along the stable directions (those whose eigenvalues are less than unity inmagnitude), and change their distance only sub-exponentially (or not at all) alongthe marginal directions (those whose eigenvalues equal unity in magnitude).

In our game of pinball the beam of neighboring trajectories is defocused alongthe unstable eigen-direction of the Jacobian matrix J.

As the heights of the strips in figure 1.9 are eectively constant, we can con-centrate on their thickness. If the height is L, then the area of the ith strip isMi Lli for a strip of width li.



Figure 1.10: The Jacobian matrix Jt maps an infinites-imal displacement x at x0 into a displacement Jt(x0)xa finite time t later.

x(t) = J t x(0)

x(0)x(0)

x(t)

Each strip i in figure 1.9 contains a periodic point xi. The finer the intervals,the smaller the variation in flow across them, so the contribution from the stripof width li is well-approximated by the contraction around the periodic point xiwithin the interval,

li = ai/|i| , (1.5)

where i is the unstable eigenvalue of the Jacobian matrix Jt(xi) evaluated atthe ith periodic point for t = Tp, the full period (due to the low dimensionality,the Jacobian can have at most one unstable eigenvalue). Only the magnitude ofthis eigenvalue matters, we can disregard its sign. The prefactors ai reflect theoverall size of the system and the particular distribution of starting values of x. Asthe asymptotic trajectories are strongly mixed by bouncing chaotically around therepeller, we expect their distribution to be insensitive to smooth variations in thedistribution of initial points.

section 19.4

To proceed with the derivation we need the hyperbolicity assumption: forlarge n the prefactors ai O(1) are overwhelmed by the exponential growth ofi, so we neglect them. If the hyperbolicity assumption is justified, we can replace section 21.1.1|Mi| Lli in (1.2) by 1/|i| and consider the sum

n =

(n)i

1/|i| ,

where the sum goes over all periodic points of period n. We now define a gener-ating function for sums over all periodic orbits of all lengths:

(z) =

n=1nz

n . (1.6)

Recall that for large n the nth level sum (1.2) tends to the limit n en, so theescape rate is determined by the smallest z = e for which (1.6) diverges:

(z)

n=1(ze)n = ze

1 ze . (1.7)



This is the property of (z) that motivated its definition. Next, we devise a formulafor (1.6) expressing the escape rate in terms of periodic orbits:

(z) =

n=1zn

(n)i|i|1

=z

|0| +z

|1| +z2

|00| +z2

|01| +z2

|10| +z2

|11|+

z3

|000| +z3

|001| +z3

|010| +z3

|100| + . . . (1.8)

For suciently small z this sum is convergent. The escape rate is now given bysection 21.3

the leading pole of (1.7), rather than by a numerical extrapolation of a sequence ofn extracted from (1.3). As any finite truncation n < ntrunc of (1.8) is a polyno-mial in z, convergent for any z, finding this pole requires that we know somethingabout n for any n, and that might be a tall order.

We could now proceed to estimate the location of the leading singularity of(z) from finite truncations of (1.8) by methods such as Pade approximants. How-ever, as we shall now show, it pays to first perform a simple resummation thatconverts this divergence into a zero of a related function.

1.5.2 Dynamical zeta function

If a trajectory retraces a prime cycle r times, its expanding eigenvalue is rp. Aprime cycle p is a single traversal of the orbit; its label is a non-repeating symbolstring of np symbols. There is only one prime cycle for each cyclic permutationclass. For example, p = 0011 = 1001 = 1100 = 0110 is prime, but 0101 = 01 is not.By the chain rule for derivatives the stability of a cycle is the same everywhere

exercise 18.2section 4.5along the orbit, so each prime cycle of length np contributes np terms to the sum

(1.8). Hence (1.8) can be rewritten as

(z) =

pnp

r=1

(znp

|p|)r=

p

nptp1 tp , tp =

znp

|p| (1.9)

where the index p runs through all distinct prime cycles. Note that we have re-summed the contribution of the cycle p to all times, so truncating the summationup to given p is not a finite time n np approximation, but an asymptotic, infinitetime estimate based by approximating stabilities of all cycles by a finite number ofthe shortest cycles and their repeats. The npznp factors in (1.9) suggest rewritingthe sum as a derivative

(z) = z ddz

pln(1 tp) .



Hence (z) is z derivative derivative of the logarithm of the infinite product

1/(z) =

p(1 tp) , tp = z

np

|p| . (1.10)

This function is called the dynamical zeta function, in analogy to the Riemannzeta function, which motivates the zeta in its definition as 1/(z). This is theprototype formula of periodic orbit theory. The zero of 1/(z) is a pole of (z),and the problem of estimating the asymptotic escape rates from finite n sumssuch as (1.2) is now reduced to a study of the zeros of the dynamical zeta function(1.10). The escape rate is related by (1.7) to a divergence of (z), and (z) diverges

section 27.1whenever 1/(z) has a zero.

section 22.4

Easy, you say: Zeros of (1.10) can be read o the formula, a zero

zp = |p|1/np

for each term in the product. Whats the problem? Dead wrong!

1.5.3 Cycle expansions

How are formulas such as (1.10) used? We start by computing the lengths andeigenvalues of the shortest cycles. This usually requires some numerical work,such as the Newton method searches for periodic solutions; we shall assume thatthe numerics are under control, and that all short cycles up to given length havebeen found. In our pinball example this can be done by elementary geometrical

chapter 16optics. It is very important not to miss any short cycles, as the calculation is asaccurate as the shortest cycle droppedincluding cycles longer than the shortestomitted does not improve the accuracy. The result of such numerics is a table ofthe shortest cycles, their periods and their stabilities.

section 33.3

Now expand the infinite product (1.10), grouping together the terms of thesame total symbol string length

1/ = (1 t0)(1 t1)(1 t10)(1 t100) = 1 t0 t1 [t10 t1t0] [(t100 t10t0) + (t101 t10t1)][(t1000 t0t100) + (t1110 t1t110)+(t1001 t1t001 t101t0 + t10t0t1)] . . . (1.11)

The virtue of the expansion is that the sum of all terms of the same total lengthchapter 23

n (grouped in brackets above) is a number that is exponentially smaller than atypical term in the sum, for geometrical reasons we explain in the next section.

section 23.1

The calculation is now straightforward. We substitute a finite set of the eigen-values and lengths of the shortest prime cycles into the cycle expansion (1.11), andobtain a polynomial approximation to 1/. We then vary z in (1.10) and determinethe escape rate by finding the smallest z = e for which (1.11) vanishes.



Figure 1.11: Approximation to a smooth dynamics(left frame) by the skeleton of periodic points, togetherwith their linearized neighborhoods, (right frame). In-dicated are segments of two 1-cycles and a 2-cyclethat alternates between the neighborhoods of the two1-cycles, shadowing first one of the two 1-cycles, andthen the other.

Figure 1.12: A longer cycle p shadowed by a pair (apseudo orbit) of shorter cycles p and p.

p

p"p

1.5.4 Shadowing

When you actually start computing this escape rate, you will find out that theconvergence is very impressive: only three input numbers (the two fixed points 0,1 and the 2-cycle 10) already yield the pinball escape rate to 3-4 significant digits!We have omitted an infinity of unstable cycles; so why does approximating the

section 23.2.2dynamics by a finite number of the shortest cycle eigenvalues work so well?

The convergence of cycle expansions of dynamical zeta functions is a conse-quence of the smoothness and analyticity of the underlying flow. Intuitively, onecan understand the convergence in terms of the geometrical picture sketched infigure 1.11; the key observation is that the long orbits are shadowed by sequencesof shorter orbits.

A typical term in (1.11) is a dierence of a long cycle {ab}minus its shadowingapproximation by shorter cycles {a} and {b} (see figure1.12),

tab tatb = tab(1 tatb/tab) = tab(1

abab) , (1.12)

where a and b are symbol sequences of the two shorter cycles. If all orbits areweighted equally (tp = znp ), such combinations cancel exactly; if orbits of similarsymbolic dynamics have similar weights, the weights in such combinations almostcancel.

This can be understood in the context of the pinball game as follows. Considerorbits 0, 1 and 01. The first corresponds to bouncing between any two disks whilethe second corresponds to bouncing successively around all three, tracing out anequilateral triangle. The cycle 01 starts at one disk, say disk 2. It then bounces



from disk 3 back to disk 2 then bounces from disk 1 back to disk 2 and so on, so itsitinerary is 2321. In terms of the bounce types shown in figure1.6, the trajectory isalternating between 0 and 1. The incoming and outgoing angles when it executesthese bounces are very close to the corresponding angles for 0 and 1 cycles. Alsothe distances traversed between bounces are similar so that the 2-cycle expandingeigenvalue 01 is close in magnitude to the product of the 1-cycle eigenvalues01.

To understand this on a more general level, try to visualize the partition ofa chaotic dynamical systems state space in terms of cycle neighborhoods as atessellation (a tiling) of the dynamical system, with smooth flow approximated byits periodic orbit skeleton, each tile centered on a periodic point, and the scaleof the tile determined by the linearization of the flow around the periodic point,as illustrated by figure 1.11.

The orbits that follow the same symbolic dynamics, such as {ab} and a pseudoorbit {a}{b} (see figure 1.12), lie close to each other in state space; long shadow-ing pairs have to start out exponentially close to beat the exponential growth inseparation with time. If the weights associated with the orbits are multiplicativealong the flow (for example, by the chain rule for products of derivatives) andthe flow is smooth, the term in parenthesis in (1.12) falls o exponentially withthe cycle length, and therefore the curvature expansions are expected to be highlyconvergent.

chapter 28

1.6 Change in time

MEN are deplorably ignorant with respect to naturalthings and modern philosophers as though dreaming in thedarkness must be aroused and taught the uses of things thedealing with things they must be made to quit the sort oflearning that comes only from books and that rests onlyon vain arguments from probability and upon conjectures.

William Gilbert, De Magnete, 1600

The above derivation of the dynamical zeta function formula for the escape ratehas one shortcoming; it estimates the fraction of survivors as a function of thenumber of pinball bounces, but the physically interesting quantity is the escaperate measured in units of continuous time. For continuous time flows, the escaperate (1.2) is generalized as follows. Define a finite state space region M suchthat a trajectory that exits M never reenters. For example, any pinball that fallsof the edge of a pinball table in figure 1.1 is gone forever. Start with a uniformdistribution of initial points. The fraction of initial x whose trajectories remainwithinM at time t is expected to decay exponentially

(t) =M dxdy (y f t(x))

M dx et .



The integral over x starts a trajectory at every x M. The integral over y testswhether this trajectory is still inM at time t. The kernel of this integral

Lt(y, x) = (y f t(x)

)(1.13)

is the Dirac delta function, as for a deterministic flow the initial point x mapsinto a unique point y at time t. For discrete time, fn(x) is the nth iterate of themap f . For continuous flows, f t(x) is the trajectory of the initial point x, and itis appropriate to express the finite time kernel Lt in terms of A, the generator ofinfinitesimal time translations

Lt = etA ,section 19.6

very much in the way the quantum evolution is generated by the Hamiltonian H,the generator of infinitesimal time quantum transformations.

As the kernel L is the key to everything that follows, we shall give it a name,and refer to it and its generalizations as the evolution operator for a d-dimensionalmap or a d-dimensional flow.

The number of periodic points increases exponentially with the cycle length(in the case at hand, as 2n). As we have already seen, this exponential proliferationof cycles is not as dangerous as it might seem; as a matter of fact, all our compu-tations will be carried out in the n limit. T

Chaos-Classical and quantum

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