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CONTENTSxiiFreddy Christiansen12.2 One-dimensional mappings .................................20012.3 Multipoint shooting method .................................201Per Dahlqvist23 Intermittency ................................................38527.3 Orbit length extremization method for billiards ...............474Carl P. Dettmann18.5 Stability ordering of cycle expansions ........................310Fotis K. Diakonos27.2 Discrete cyclists relaxation methods .........................470G. Bard ErmentroutExercise 5.1Mitchell J. FeigenbaumAppendix B.3: Symplectic invariance ............................659Jonathan HalcrowExample 3.5: Sections of Lorenz flow .............................52Example 4.6: Stability of Lorenz flow equilibria ....................72Example 4.7: Lorenz flow: Global portrait .........................74Example 9.2: Desymmetrization of Lorenz flow ...................134Example 10.5: Lorenz flow: a 1−d return map .....................159Exercises 9.8, 9.9 and figure 2.4Kai T. Hansen10.2.1 Unimodal map symbolic dynamics .........................15813.6 Topological zeta function for an infinite partition ..............22610.3 Kneading theory ...........................................163figures throughout the textRainer KlagesFigure 24.5Yueheng LanSolutions 1.1, 2.1, 2.2, 2.3, 2.4, 2.5, 9.2, 9.1, 10.6, 14.1, 14.2, 14.3, 14.5,14.7, 14.10, 15.1 and figures 1.9, 9.1, 9.6 10.3,Bo LiSolutions 30.2, 30.1, 31.1Joachim Mathiesen15.3 Lyapunov exponents .......................................263Rössler flow figures, tables, cycles in sections 15.3, ?? and exercise 12.7Yamato MatsuokaFigure 11.3Rytis Paškauskas
CONTENTSxiii4.5.1 Stability of Poincaré return maps .............................785.3 Stability of Poincaré map cycles ...............................89Exercises 2.8, 3.1, 4.4 and solution 4.1Adam Prügel-BennetSolutions 1.2, 2.10, 8.1, 17.1, 18.2 21.3, 27.1,Lamberto Rondoni14 Transporting densities ........................................23512.1.1 Cycles from long time series ..............................19720.2.1 Unstable periodic orbits are dense .........................339Juri RolfSolution 21.3Per E. Rosenqvistexercises, figures throughout the textHans Henrik Rugh21 Why does it work? .......................................... 347Peter Schmelcher27.2 Discrete cyclists relaxation methods .........................470Evangelos SiminosExample 3.5: Sections of Lorenz flow .............................52Example 4.6: Stability of Lorenz flow equilibria ....................72Example 4.7: Lorenz flow: Global portrait .........................74Example 9.2: Desymmetrization of Lorenz flow ...................134Example 10.5: Lorenz flow: a 1−d return map .....................159Exercises 9.8, 9.9Gábor SimonRössler flow figures, tables, cycles in chapters 2, 12 and exercise 12.7Edward A. Spiegel2 Flows .........................................................3214 Transporting densities ........................................235Luz V. Vela-Arevalo7.1 Hamiltonian flows .......................................... 108Exercises 7.1, 7.3, 7.5
Page 1 and 2: . Chaos: Classical and QuantumI: De
Page 3 and 4: CONTENTSiii5.3 Stability of Poincar
Page 5 and 6: CONTENTSv18 Cycle expansions 29918.
Page 7 and 8: CONTENTSviiPart II: Quantum chaos29
Page 9 and 10: CONTENTSixPart III: Appendices on C
Page 11: CONTENTSxiContributorsNo man but a
Page 16 and 17: Chapter 1OvertureIf I have seen les
Page 18 and 19: CHAPTER 1. OVERTURE 3this material,
Page 20 and 21: CHAPTER 1. OVERTURE 5231323212Figur
Page 22 and 23: CHAPTER 1. OVERTURE 7Figure 1.4: Dy
Page 24 and 25: CHAPTER 1. OVERTURE 9Figure 1.5: Ka
Page 26 and 27: CHAPTER 1. OVERTURE 11Figure 1.7: T
Page 28 and 29: CHAPTER 1. OVERTURE 13of survivors
Page 30 and 31: CHAPTER 1. OVERTURE 15where Λ i is
Page 32 and 33: CHAPTER 1. OVERTURE 171.5.3 Cycle e
Page 34 and 35: CHAPTER 1. OVERTURE 191.6 Evolution
Page 36 and 37: CHAPTER 1. OVERTURE 21The beauty of
Page 38 and 39: CHAPTER 1. OVERTURE 23So, no proofs
Page 40 and 41: CHAPTER 1. OVERTURE 25But the power
Page 42 and 43: CHAPTER 1. OVERTURE 27measures of
Page 44 and 45: EXERCISES 29Exercises1.1. 3-disk sy
Page 46 and 47: References 31[1.29] D. Ruelle, “F
Page 48 and 49: CHAPTER 2. GO WITH THE FLOW 33Figur
Page 50 and 51: CHAPTER 2. GO WITH THE FLOW 35very
Page 52 and 53: CHAPTER 2. GO WITH THE FLOW 37asymp
Page 54 and 55: CHAPTER 2. GO WITH THE FLOW 39Z(t)3
Page 56 and 57: CHAPTER 2. GO WITH THE FLOW 41some
Page 58 and 59: EXERCISES 43The exercises that you
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CHAPTER 3. DISCRETE TIME DYNAMICS 4
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EXERCISES 59Exercises3.1. Poincaré
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Chapter 4Local stability(R. Mainier
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CHAPTER 4. LOCAL STABILITY 63x(t)δ
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CHAPTER 4. LOCAL STABILITY 65Hence
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CHAPTER 4. LOCAL STABILITY 67Exampl
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CHAPTER 4. LOCAL STABILITY 69are fl
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CHAPTER 4. LOCAL STABILITY 714.3 St
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CHAPTER 4. LOCAL STABILITY 73Figure
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CHAPTER 4. LOCAL STABILITY 75task a
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CHAPTER 4. LOCAL STABILITY 77hand s
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CHAPTER 4. LOCAL STABILITY 79Figure
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EXERCISES 81The nomenclature tends
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Chapter 5Cycle stabilityTopological
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CHAPTER 5. CYCLE STABILITY 85Figure
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CHAPTER 5. CYCLE STABILITY 87Exampl
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CHAPTER 5. CYCLE STABILITY 89so a p
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EXERCISES 91infinite set of metric
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Chapter 6Get straightWe owe it to a
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CHAPTER 6. GET STRAIGHT 95smoothly,
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CHAPTER 6. GET STRAIGHT 97Figure 6.
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CHAPTER 6. GET STRAIGHT 99Example 6
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CHAPTER 6. GET STRAIGHT 1016.4 Rect
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CHAPTER 6. GET STRAIGHT 1036.6 Cycl
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CHAPTER 6. GET STRAIGHT 105Remark 6
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References 107[6.10] I. Percival an
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CHAPTER 7. HAMILTONIAN DYNAMICS 109
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EXERCISES 117Put your right hand on
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References 119[7.5] D.G. Sterling,
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CHAPTER 8. BILLIARDS 121Figure 8.1:
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CHAPTER 8. BILLIARDS 123Consider a
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EXERCISES 125CommentaryRemark 8.1 B
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References 127[8.7] G. Vattay, A. W
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CHAPTER 9. WORLD IN A MIRROR 1299.1
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CHAPTER 9. WORLD IN A MIRROR 1319.1
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CHAPTER 9. WORLD IN A MIRROR 133{0}
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CHAPTER 9. WORLD IN A MIRROR 135Fig
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CHAPTER 9. WORLD IN A MIRROR 141the
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CHAPTER 9. WORLD IN A MIRROR 143Exa
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CHAPTER 9. WORLD IN A MIRROR 145(1)
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EXERCISES 147Exercises9.1. 3-disk f
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REFERENCES 1499.9. Proto-Lorenz sys
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References 151[9.28] R. Miranda and
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CHAPTER 10. QUALITATIVE DYNAMICS, F
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EXERCISES 171Remark 10.2 Counting p
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References 173[10.10] A. Boyarski,
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REFERENCES 191(a) (easy) Consider a
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References 193[11.26] K.T. Hansen,
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Chapter 12Fixed points, and how to
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CHAPTER 12. FIXED POINTS, AND HOW T
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EXERCISES 209Exercises12.1. Cycles
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References 211[12.11] B. Doyon and
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CHAPTER 13. COUNTING 2133e n ln 2
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CHAPTER 13. COUNTING 21513.2 Topolo
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CHAPTER 13. COUNTING 217We could no
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CHAPTER 13. COUNTING 219Figure 13.1
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CHAPTER 13. COUNTING 221terms of cy
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CHAPTER 13. COUNTING 223Example 13.
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Table 13.3: List of the 3-disk prim
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CHAPTER 13. COUNTING 227Figure 13.4
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CHAPTER 13. COUNTING 229step in der
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EXERCISES 231the corresponding tran
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REFERENCES 233As the rule does not
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Chapter 14Transporting densitiesPau
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CHAPTER 14. TRANSPORTING DENSITIES
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EXERCISES 251quence of Gaussians∫
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References 253[14.13] G. Froyland,
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CHAPTER 15. AVERAGING 25515.1.1 Tim
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CHAPTER 15. AVERAGING 257of the dyn
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CHAPTER 15. AVERAGING 259so if we c
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CHAPTER 15. AVERAGING 261Figure 15.
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CHAPTER 15. AVERAGING 263δx0x(0)δ
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CHAPTER 15. AVERAGING 265Figure 15.
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CHAPTER 15. AVERAGING 267Here we ha
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REFERENCES 26915.1. How unstable is
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Chapter 16Trace formulasThe trace f
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CHAPTER 16. TRACE FORMULAS 273magni
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CHAPTER 16. TRACE FORMULAS 275This
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CHAPTER 16. TRACE FORMULAS 277be th
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CHAPTER 16. TRACE FORMULAS 279the f
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EXERCISES 281Now that we have a tra
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Chapter 17Spectral determinants“I
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CHAPTER 17. SPECTRAL DETERMINANTS 2
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REFERENCES 297(d) Let p be all prim
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Chapter 18Cycle expansionsRecycle..
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CHAPTER 18. CYCLE EXPANSIONS 301We
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CHAPTER 18. CYCLE EXPANSIONS 303For
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CHAPTER 18. CYCLE EXPANSIONS 305Giv
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CHAPTER 18. CYCLE EXPANSIONS 307and
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CHAPTER 18. CYCLE EXPANSIONS 30918.
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CHAPTER 18. CYCLE EXPANSIONS 311whi
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CHAPTER 18. CYCLE EXPANSIONS 313Fig
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CHAPTER 18. CYCLE EXPANSIONS 315fea
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EXERCISES 317Exercises18.1. Cycle e
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REFERENCES 31918.12. Ulam map is co
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CHAPTER 19. DISCRETE FACTORIZATION
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REFERENCES 335tonian∑H(ɛ) = −J
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CHAPTER 20. WHY CYCLE? 337get thinn
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CHAPTER 20. WHY CYCLE? 339of the it
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CHAPTER 20. WHY CYCLE? 341As is usu
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CHAPTER 20. WHY CYCLE? 3431. due to
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EXERCISES 345connected due to the n
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Chapter 21Why does it work?Bloch:
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CHAPTER 21. WHY DOES IT WORK? 349in
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CHAPTER 21. WHY DOES IT WORK? 351is
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CHAPTER 21. WHY DOES IT WORK? 353fi
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CHAPTER 21. WHY DOES IT WORK? 355wo
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CHAPTER 21. WHY DOES IT WORK? 357Th
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CHAPTER 21. WHY DOES IT WORK? 359As
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CHAPTER 21. WHY DOES IT WORK? 3611.
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CHAPTER 21. WHY DOES IT WORK? 363Fi
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CHAPTER 21. WHY DOES IT WORK? 365Ex
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CHAPTER 21. WHY DOES IT WORK? 367R
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CHAPTER 21. WHY DOES IT WORK? 369we
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References 371[21.7] D. Ruelle, “
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Chapter 22Thermodynamic formalismBe
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CHAPTER 22. THERMODYNAMIC FORMALISM
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REFERENCES 383tory. Biham and Kvale
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Chapter 23IntermittencySometimes Th
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CHAPTER 23. INTERMITTENCY 38710.8f(
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CHAPTER 23. INTERMITTENCY 3891a0.8F
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CHAPTER 23. INTERMITTENCY 391It may
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CHAPTER 23. INTERMITTENCY 393to arb
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CHAPTER 23. INTERMITTENCY 395where
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CHAPTER 23. INTERMITTENCY 397assumi
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CHAPTER 23. INTERMITTENCY 399Figure
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CHAPTER 23. INTERMITTENCY 401Table
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CHAPTER 23. INTERMITTENCY 403for al
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CHAPTER 23. INTERMITTENCY 405Changi
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CHAPTER 23. INTERMITTENCY 407∫1/
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CHAPTER 23. INTERMITTENCY 409with i
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REFERENCES 411This leads to the fol
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Chapter 24Deterministic diffusionTh
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CHAPTER 24. DETERMINISTIC DIFFUSION
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References 433[24.7] P. Cvitanović
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CHAPTER 25. TURBULENCE? 43525.1 Flu
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CHAPTER 25. TURBULENCE? 4372vt, t),
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CHAPTER 25. TURBULENCE? 439Figure 2
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CHAPTER 25. TURBULENCE? 441The equi
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CHAPTER 25. TURBULENCE? 44310Figure
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CHAPTER 25. TURBULENCE? 445Table 25
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CHAPTER 25. TURBULENCE? 447The mean
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CHAPTER 25. TURBULENCE? 449(u x ) 2
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REFERENCES 451for finite dimensiona
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Chapter 26NoiseHe who establishes h
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CHAPTER 26. NOISE 45526.2 Brownian
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CHAPTER 26. NOISE 457The left hand
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CHAPTER 26. NOISE 459in time t is g
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EXERCISES 461Exercises26.1. Who ord
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References 463[26.32] O. Cepas and
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CHAPTER 27. RELAXATION FOR CYCLISTS
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Table 27.3: All prime cycles up to
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EXERCISES 477Exercises27.1. Evaluat
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Chapter 28Irrationally windingI don
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CHAPTER 28. IRRATIONALLY WINDING 48
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REFERENCES 497References[28.1] P. C
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Chaos: Classical and QuantumVolume
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CHAPTER 29. PROLOGUE 50129.1 Quantu
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CHAPTER 29. PROLOGUE 5031086r 24Fig
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REFERENCES 505References[29.1] M. B
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CHAPTER 30. QUANTUM MECHANICS, BRIE
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CHAPTER 30. QUANTUM MECHANICS, BRIE
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Chapter 31WKB quantizationThe wave
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CHAPTER 31. WKB QUANTIZATION 513Fig
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CHAPTER 31. WKB QUANTIZATION 51531.
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CHAPTER 31. WKB QUANTIZATION 517Fig
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EXERCISES 519soft potential. The co
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Chapter 32Semiclassical evolutionWi
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CHAPTER 32. SEMICLASSICAL EVOLUTION
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REFERENCES 543References[32.1] A. E
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CHAPTER 33. SEMICLASSICAL QUANTIZAT
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EXERCISES 555Exercises33.1. Monodro
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Chapter 34Quantum scatteringScatter
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CHAPTER 34. QUANTUM SCATTERING 559F
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CHAPTER 34. QUANTUM SCATTERING 561T
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CHAPTER 34. QUANTUM SCATTERING 563t
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CHAPTER 34. QUANTUM SCATTERING 565W
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EXERCISES 567CommentaryRemark 34.1
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References 569[34.13] J.S. Faulkner
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Chapter 35Chaotic multiscattering(A
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CHAPTER 35. CHAOTIC MULTISCATTERING
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CHAPTER 36. HELIUM ATOM 591eeFigure
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CHAPTER 36. HELIUM ATOM 593765r2432
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CHAPTER 36. HELIUM ATOM 59561010014
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CHAPTER 36. HELIUM ATOM 597with H Q
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CHAPTER 36. HELIUM ATOM 599S p = 2T
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CHAPTER 36. HELIUM ATOM 6010-0.5N=5
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CHAPTER 36. HELIUM ATOM 603and m p
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CHAPTER 36. HELIUM ATOM 605N n j =
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CHAPTER 36. HELIUM ATOM 607Remark 3
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REFERENCES 609displacements along t
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CHAPTER 37. DIFFRACTION DISTRACTION
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EXERCISES 623Exercises37.1. Station
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EpilogueNowadays, whatever the trut
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References 627of “structural stab
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Indexaction, 232admissibletrajector
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INDEX 631matrix, 54, 124Gatto Nerop
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INDEX 633Poincaré section, 41-42,
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INDEX 635unimodal, 140winding numbe
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Appendix AA brief history of chaosL
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APPENDIX A. A BRIEF HISTORY OF CHAO
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REFERENCES 6493. Hénon case: The f
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Appendix BLinear stabilityMopping u
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APPENDIX B. LINEAR STABILITY 653(a)
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APPENDIX B. LINEAR STABILITY 655Usi
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APPENDIX B. LINEAR STABILITY 657Deg
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APPENDIX B. LINEAR STABILITY 659i.e
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EXERCISES 661ExercisesB.1. Real rep
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APPENDIX C. IMPLEMENTING EVOLUTION
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APPENDIX C. IMPLEMENTING EVOLUTION
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EXERCISES 667Note that the billiard
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Appendix DSymbolic dynamics techniq
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APPENDIX D. SYMBOLIC DYNAMICS TECHN
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Appendix ECounting itinerariesE.1 C
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Appendix FFinding cycles(C. Chandre
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APPENDIX F. FINDING CYCLES 6852.521
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Appendix GTransport of vector field
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APPENDIX G. TRANSPORT OF VECTOR FIE
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EXERCISES 695for the zeta-functionF
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References 697[G.17] M.J. Feigenbau
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APPENDIX H. DISCRETE SYMMETRIES OF
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λ 3 − λ 2λ 3 − λ 2. ..⎞
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REFERENCES 725The point of the abov
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APPENDIX I. CONVERGENCE OF SPECTRAL
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Appendix JInfinite dimensional oper
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APPENDIX J. INFINITE DIMENSIONAL OP
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EXERCISES 745such that the spectral
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Appendix KStatistical mechanics rec
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APPENDIX K. STATISTICAL MECHANICS R
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f fff fff fff fffffffffff fff fff f
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REFERENCES 769space H) such that fo
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References 771[K.20] M. E. Fisher.
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APPENDIX L. NOISE/QUANTUM CORRECTIO
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REFERENCES 785Figure L.2: A typical
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References 787[L.16] A. Wirzba, CHA
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APPENDIX T. PROJECTS 8655. or evalu
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APPENDIX T. PROJECTS 867Figure T.1:
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APPENDIX T. PROJECTS 869Figure T.2:
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REFERENCES 871for larger Λ more po
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References 873figure 24.4 Λ D414(a
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