DIRECTIONS IN CHAOS VOL. 7 APPLIED SYMBOLIC DYNAMICS AND CHAOS Bai-Lin Hao Wei-Mou Zheng The Institute of Theoretical Physics Academia Sinica, China Vfö World Scientific wl Singapore Sinaaoore NewJersev Jersey L London Hong Kong
Contents Preface xiii 1 Introduction 1 1.1 Dynamical Systems 2 1.1.1 Phase Space and Orbits 2 1.1.2 Parameters and Bifurcation of Dynamical Behavior 2 1.1.3 Examples of Dynamical Systems 3 1.2 Symbolic dynamics as Coarse-Grained Description of Dynamics... 5 1.2.1 Fine-Grained and Coarse-Grained Descriptions 6 1.2.2 Symbolic Dynamics as the Simplest Dynamics 6 1.3 Abstract versus Applied Symbolic Dynamics 8 1.3.1 Abstract Symbolic Dynamics 8 1.3.2 Applied Symbolic Dynamics 9 1.4 Literature on Symbolic Dynamics 11 2 Symbolic Dynamics of Unimodal Maps 13 2.1 Symbolic Sequences in Unimodal Maps 15 2.1.1 Numerical Orbit and Symbolic Sequence 15 2.1.2 Symbolic Sequence and Functional Composition 21 2.1.3 The Word-Lifting Technique 22 2.2 The Quadratic Map 24 2.2.1 An Over-Simplified Population Model 24 2.2.2 Bifurcation Diagram of the Quadratic Map 26 2.2.3 Dark Lines in the Bifurcation Diagram 30 2.3 Ordering of Symbolic Sequences and the Admissibility Condition... 37 2.3.1 Property of Monotone Functions 38 2.3.2 The Ordering Rule 38 2.3.3 Dynamical Invariant Range and Kneading Sequence 42 2.3.4 The Admissibility Condition 43 2.4 The Periodic Window Theorem 45 2.4.1 Periodic Window Theorem 46 2.4.2 Construction of Median Words 49 2.4.3 The MSS Table of Kneading Sequences 50 2.4.4 Nomenclature of Unstable Periodic Orbits 52 vii
viii Contents 2.5 Composition Rules 53 2.5.1 The *-Composition 53 2.5.2 Generalized Composition Rule 57 2.5.3 Proof of the Generalized Composition Rule 60 2.5.4 Applications of the Generalized Composition Rule 62 2.5.5 Further Remarks on Composition Rules 65 2.6 Coarse-Grained Chaos 67 2.6.1 Chaos in the Surjective Unimodal Map 68 2.6.2 Chaos in px 00 Maps 74 2.7 Topological Entropy 82 2.8 Piecewise Linear Maps and Metrie Representation of Symbolic Sequences 83 2.8.1 The Tent Map and Shift Map 84 2.8.2 The A-Expansion of Real Numbers 85 2.8.3 Characteristic Function of the Kneading Sequence 86 2.8.4 Mapping of Subintervals and the Stefan Matrix 86 2.8.5 Markov Partitions and Generating Partitions 93 2.8.6 Metrie Representation of Symbolic Sequences 96 2.8.7 Piecewise Linear Expanding Map 100 3 Maps with Multiple Critical Points 103 3.1 General Discussion 104 3.1.1 The Ordering Rule 105 3.1.2 Admissibility and Compatibility of Kneading Sequences.... 106 3.2 The Antisymmetric Cubic Map 106 3.2.1 Symbolic Sequences and Their Ordering 109 3.2.2 Admissibility Conditions 110 3.2.3 Generation of Superstable Median Words 112 3.3 Symmetry Breaking and Restoration 118 3.3.1 Symmetry Breaking of Symmetrie Orbits 120 3.3.2 Analysis of Symmetry Restoration 122 3.4 The Gap Map 125 3.4.1 The Kneading Plane 127 3.4.2 Contacts of Even-Odd Type 131 3.4.3 Self-Similar Structure in the Kneading Plane 132 3.4.4 Criterion for Topological Chaos 135 3.5 The Lorenz-Like Map 138 3.5.1 Ordering Rule and Admissibility Conditions 139 3.5.2 Construction of the Kneading Plane 139 3.5.3 Contacts and Intersections 140 3.5.4 Farey and Doubling Transformations 141 3.6 General Cubic Maps 142 3.6.1 Skeleton, Bones and Joints in Kneading Plane 145 3.6.2 The Construction of the Kneading Plane 147 3.6.3 The *-Composition Rules 151
Contents ix 3.6.4 The (-,+,-) Type Cubic Map 152 3.7 The Sine-Square Map 156 3.7.1 Symbolic Sequences and Word-Lifting Technique 157 3.7.2 Ordering Rule and Admissibility Conditions 159 3.7.3 Generation of Kneading Sequences 160 3.7.4 Joints and Bones in the Kneading Plane 161 3.7.5 Skeleton of Superstable Orbits and Existence of Topological Chaos 165 3.8 The Lorenz-Sparrow Maps 166 3.8.1 Ordering and Admissibility of Symbolic Sequences 167 3.8.2 Generation of Compatible Kneading Pairs 169 3.8.3 Generation of Admissible Sequences for Given Kneading Pair. 170 3.8.4 Metrie Representation of Symbolic Sequences 172 3.8.5 One-Parameter Limits of Lorenz-Sparrow Maps 173 3.9 Piecewise Linear Maps 174 3.9.1 Piecewise Linear Maps with Multiple Critical Points 174 3.9.2 Kneading Determinants 175 4 Symbolic Dynamics of Circle Maps 177 4.1 The Physics of Linear and Nonlinear Oscillators 178 4.2 Circle Maps and Their Lifts 179 4.2.1 The Rigid Rotation Bare Circle Map 180 4.2.2 The Sine-Circle Map 182 4.2.3 Lift of Circle Maps 183 4.2.4 Rotation Number and Rotation Interval 184 4.2.5 Arnold Tongues in the Parameter Plane 186 4.3 Continued Fractions and Farey Tree 187 4.3.1 Farey Tree: Rational Fraction Representation 187 4.3.2 Farey Tree: Continued Fraction Representation 188 4.3.3 Farey Tree: Farey Addresses and Farey Matrices 191 4.3.4 More on Continued Fraction and Farey Representations... 193 4.3.5 Farey Tree: Symbolic Representation 197 4.4 Farey Transformations and Well-Ordered Orbits 200 4.4.1 Well-Ordered Symbolic Sequences 201 4.4.2 Farey Transformations as Composition Rules 201 4.3.3 Extreme Property of Well-Ordered Periodic Sequences... 202 4.4.4 Generation of R and L 205 III dx II1111 4.5 Circle Map with Non-Monotone Lift 207 4.5.1 Symbolic Sequences and Their Continuous Transformations.. 207 4.5.2 Ordering Rule and Admissibility Condition 208 4.5.3 Existence of Well-Ordered Symbolic Sequences 209 4.5.4 The Farey Transformations 210 4.5.5 Existence of Symbolic Sequence without Rotation Number.. 211 4.6 Kneading Plane of Circle Maps 212
x Contents 4.6.1 Arnold Tongue with Rotation Number 1/2 212 4.6.2 Doubly Superstable Kneading Sequences: Joints and Bones.. 213 4.6.3 Generation of Kneading Sequences K g and K s 215 4.6.4 Construction of the Kneading Plane 217 4.7 Piecewise Linear Circle Maps and Topological Entropy 218 4.7.1 The Sawtooth Circle Map 218 4.7.2 Circle Map with Given Kneading Sequences 219 4.7.3 Kneading Determinant and Topological Entropy 221 4.7.4 Construction of a Map from a Given Kneading Sequence... 222 4.7.5 Rotation Interval and Well-Ordered Periodic Sequences... 222 5 Symbolic Dynamics of Two-Dimensional Maps 225 5.1 General Discussion 227 5.1.1 Bi-Infinite Symbolic Sequences 227 5.1.2 Decomposition of the Phase Plane. 229 5.1.3 Tangencies and Admissibility Conditions 230 5.1.4 Admissibility Conditions in Symbolic Plane 231 5.2 Invariant Manifolds and Dynamical Foliations of Phase Plane 233 5.2.1 Stable and Unstable Invariant Manifolds 233 5.2.2 Dynamical Foliations of the Phase Plane 236 5.2.3 Summary and Discussion 238 5.3 The Tel Map 240 5.3.1 Forward and Backward Symbolic Sequences 241 5.3.2 Dynamical Foliations of Phase Space and Their Ordering... 242 5.3.3 Forbidden and Allowed Zones in Symbolic Plane 249 5.3.4 The Admissibility Conditions 252 5.3.5 Summary 255 5.4 The Lozi Map 256 5.4.1 Forward and Backward Symbolic Sequences 258 5.4.2 Dynamical Foliations of the Phase Space 259 5.4.3 Ordering of the Forward and Backward Foliations... 267 5.4.4 Allowed and Forbidden Zones in Symbolic Plane 269 5.4.5 Discussion of the Admissibility Condition 273 5.5 The Henon Map 275 5.5.1 Fixed Points and Their Stability 276 5.5.2 Determination of Partition Lines in Phase Plane 278 5.5.3 Henon-Type Symbolic Dynamics 282 5.5.4 Symbolic Analysis at Typical Parameter Values 283 5.6 The Dissipative Standard Map 287 5.6.1 Dynamical Foliations of the Phase Plane 287 5.6.2 Ordering of Symbolic Sequences 289 5.6.3 Symbolic Plane and Admissibility of Symbolic Sequences... 291 5.7 The Stadium Billiard Problem 294 5.7.1 A Coding Based on Lifting 295
Contents XI 5.7.2 Relation to Other Codings 298 5.7.3 The Half-Stadium 300 5.7.4 Summary 301 6 Application to Ordinary Differential Equations 303 6.1 General Discussion 305 6.1.1 Three Types of ODEs 305 6.1.2 On Numerical Integration of Differential Equations 306 6.1.3 Numerical Calculation of the Poincare Maps 308 6.2 The Periodically Forced Brusselator 312 6.2.1 The Brusselator Viewed from The Standard Map 314 6.2.2 Transition from Annular to Interval Dynamics 317 6.2.3 Symbolic Analysis of Interval Dynamics 321 6.3 The Lorenz Equations 326 6.3.1 Summary of Known Properties 328 6.3.2 Construction of Poincare and Return Maps 330 6.3.3 One-Dimensional Symbolic Dynamics Analysis 334 6.3.4 Symbolic Dynamics of the 2D Poincare Maps 337 6.3.5 Stable Periodic Orbits 345 6.3.6 Concluding Remarks 352 6.4 Summary of Other ODE Systems 352 6.4.1 The Driven Two-Well Duffing Equation 353 6.4.2 The NMR-Laser Model 354 7 Counting the Number of Periodic Orbits 355 7.1 Periodic versus Chaotic Regimes 355 7.1.1 Stable Versus Unstable Periods in 1D Maps. 356 7.1.2 Notations and Summary of Results 358 7.1.3 A Few Number Theory Notations and Functions 361 7.2 Number of Periodic Orbits in a Class of One-Parameter Maps... 362 7.2.1 Number of Admissible Words in Symbolic Dynamics 362 7.2.2 Number of Tangent and Period-Doubling Bifurcations 363 7.2.3 Recursion Formula for the Total Number of Periods 365 7.2.4 Symmetry Types of Periodic Sequences 366 7.2.5 Explicit Solutions to the Recurrence Relations 370 7.2.6 Finite Lambda Auto-Expansion of Real Numbers 371 7.3 Other Aspects of the Counting Problem 373 7.3.1 The Number of Roots of the "Dark Line" Equation 373 7.3.2 Number of Saddle Nodes in Forming Smale Horseshoe 373 7.3.3 Number of Solutions of Renormalization Group Equations.. 374 7.4 Counting Formulae for General Continuous Maps 375 7.5 Number of Periods in Maps With Discontinuity 377 7.5.1 Number of Periods in the Gap Map 377 7.5.2 Number of Periods in the Lorenz-Like Map 379
xii Contents 7.6 Summary of the Counting Problem 381 7.7 Cycle Expansion for Topological Entropy 381 8 Symbolic Dynamics and Grammatical Complexity 385 8.1 Formal Languages and Their Complexity 386 8.1.1 Formal Language 386 8.1.2 Chomsky Hierarchy of Grammatical Complexity 388 8.1.3 The L-System 389 8.2 Regulär Language and Finite Automaton 390 8.2.1 Finite Automaton 390 8.2.2 Regulär Language 391 8.2.3 Stefan Matrix as Transfer Function for Automaton 391 8.3 Beyond Regulär Languages 395 8.3.1 Feigenbaum and Generalized Feigenbaum Limiting Sets... 396 8.3.2 Even and Odd Fibonacci Sequences 396 8.3.3 Odd Maximal Primitive Prefixes and Kneading Map 398 8.3.4 Even Maximal Primitive Prefixes and Distinct Excluded Blocks 401 8.4 Summary of Results 402 9 Symbolic Dynamics and Knot Theory 403 9.1 Knots and Links 404 9.2 Knots and Links from Unimodal Maps 406 9.3 Linking Numbers 410 9.4 Discussion 411 Appendix 413 A.l Program to Generate Admissible Sequences 413 A.2 Program to Draw Dynamical Foliations of a 2D Map 419 References 423 R.l Books 423 R.2 Papers 424 Subject Index 439