实用符号动力学与混沌：英文版=Applide Symbolic Dynamic and Chaos【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

实用符号动力学与混沌：英文版=Applide Symbolic Dynamic and ChaosPDF格式电子书版下载

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1 Introduction1

1.1 Dynamical Systems2

1.1.1 Phase Space and Orbits2

1.1.2 Parameters and Bifurcation of Dynamical Behavior3

1.1.3 Examples of Dynamical Systems3

1.2 Symbolic Dynamics as Coarse-Grained Description of Dynamics6

1.2.1 Fine-Grained and Coarse-Grained Descriptions7

1.2.2 Symbolic Dynamics as the Simplest Dynamics8

1.3 Abstract versus Applied Symbolic Dynamics10

1.3.1 Abstract Symbolic Dynamics10

1.3.2 Applied Symbolic Dynamics11

1.4 Literature on Symbolic Dynamics13

2 Symbolic Dynamics of Unimodal Maps16

2.1 Symbolic Sequences in Unimodal Maps18

2.1.1 Numerical Orbit and Symbolic Sequence19

2.1.2 Symbolic Sequence and Functional Composition26

2.1.3 The Word-Lifting Technique27

2.2 The Quadratic Map29

2.2.1 An Over-Simplified Population Model29

2.2.2 Bifurcation Diagram of the Quadratic Map31

2.2.3 Dark Lines in the Bifurcation Diagram37

2.3 Ordering of Symbolic Sequences and the Admissibility Condition43

2.3.1 Property of Monotone Functions44

2.3.2 The Ordering Rule45

2.3.3 Dynamical Invariant Range and Kneading Sequence49

2.3.4 The Admissibility Condition51

2.4 The Periodic Window Theorem54

2.4.1 The Periodic Window Theorem54

2.4.2 Construction of Median Words57

2.4.3 The MSS Table of Kneading Sequences59

2.4.4 Nomenclature of Unstable Periodic Orbits61

2.5 Composition Rules62

2.5.1 The*-Composition63

2.5.2 Generalized Composition Rule67

2.5.3 Proof of the Generalized Composition Rule70

2.5.4 Applications of the Generalized Composition Rule73

2.5.5 Further Remarks on Composition Rules77

2.6 Coarse-Grained Chaos79

2.6.1 Chaos in the Surjective Unimodal Map80

2.6.2 Chaos in ρλ∞Maps87

2.7 Topological Entropy95

2.8 Piecewise Linear Maps and Metric Representation of Symbolic Sequences97

2.8.1 The Tent Map and Shift Map97

2.8.2 Theλ-Expansion of Real Numbers98

2.8.3 Characteristic Function of the Kneading Sequence99

2.8.4 Mapping of Subintervals and the Stefan Matrix100

2.8.5 Markov Partitions and Generating Partitions107

2.8.6 Metric Representation of Symbolic Sequences111

2.8.7 Piecewise Linear Expanding Map115

3 Maps with Multiple Critical Points117

3.1 General Discussion119

3.1.1 The Ordering Rule120

3.1.2 Construction of a Map from a Given Kneading Sequence120

3.2 The Antisymmetric Cubic Map121

3.2.1 Symbolic Sequences and Their Ordering124

3.2.2 Admissibility Conditions125

3.2.3 Generation of Superstable Median Words129

3.3 Symmetry Breaking and Restoration135

3.3.1 Symmetry Breaking of Symmetric Orbits137

3.3.2 Analysis of Symmetry Restoration140

3.4 The Gap Map143

3.4.1 The Kneading Plane146

3.4.2 Contacts of Even-Odd Type149

3.4.3 Self-Similar Structure in the Kneading Plane151

3.4.4 Criterion for Topological Chaos155

3.5 The Lorenz-Like Map159

3.5.1 Ordering Rule and Admissibility Conditions159

3.5.2 Construction of the Kneading Plane160

3.5.3 Contacts and Intersections161

3.5.4 Farey and Doubling Transformations162

3.6 General Cubic Maps163

3.6.1 Skeleton，Bones and Joints in Kneading Plane165

3.6.2 The Construction of the Kneading Plane169

3.6.3 The*-Composition Rules173

3.6.4 The Down-Up-Down Type Cubic Map174

3.7 The Sine-Square Map178

3.7.1 Symbolic Sequences and Word-Lifting Technique179

3.7.2 Ordering Rule and Admissibility Conditions181

3.7.3 Generation of Kneading Sequences182

3.7.4 Joints and Bones in the Kneading Plane184

3.7.5 Skeleton of Superstable Orbits and Existence of Topological Chaos188

3.8 The Lorenz-Sparrow Maps190

3.8.1 Ordering and Admissibility of Symbolic Sequences190

3.8.2 Generation of Compatible Kneading Pairs193

3.8.3 Generation of Admissible Sequences for a Given Kneading Pair194

3.8.4 Metric Representation of Symbolic Sequences196

3.8.5 One-Parameter Limits of Lorenz-Sparrow Maps197

3.9 Piecewise Linear Maps198

3.9.1 Piecewise Linear Maps with Multiple Critical Points199

3.9.2 Kneading Determinants200

4 Symbolic Dynamics of Circle Maps203

4.1 The Physics of Linear and Nonlinear Oscillators205

4.2 Circle Maps and Their Lifts206

4.2.1 The Rigid Rotation—Bare Circle Map207

4.2.2 The Sine-Circle Map209

4.2.3 Lift of Circle Maps210

4.2.4 Rotation Number and Rotation Interval212

4.2.5 Arnold Tongues in the Parameter Plane214

4.3 Continued Fractions and Farey Tree215

4.3.1 Farey Tree：Rational Fraction Representation215

4.3.2 Farey Tree：Continued Fraction Representation216

4.3.3 Farey Tree：Farey Addresses and Farey Matrices219

4.3.4 More on Continued Fractions and Farey Representations222

4.3.5 Farey Tree：Symbolic Representation226

4.4 Farey Transformations and Well-Ordered Orbits229

4.4.1 Well-Ordered Symbolic Sequences230

4.4.2 Farey Transformations as Composition Rules231

4.4.3 Extreme Property of Well-Ordered Periodic Sequences232

4.4.4 Generation of R°max and L°min235

4.5 Circle Map with Non-Monotone Lift238

4.5.1 Symbolic Sequences and Their Continuous Transformations238

4.5.2 Ordering Rule and Admissibility Condition240

4.5.3 Existence of Well-Ordered Symbolic Sequences240

4.5.4 The Farey Transformations242

4.5.5 Existence of Symbolic Sequence without Rotation Number243

4.6 Kneading Plane of Circle Maps244

4.6.1 Arnold Tongue with Rotation Number 1/2244

4.6.2 Doubly Superstable Kneading Sequences：Joints and Bones246

4.6.3 Generation of Kneading Sequences Kg and Ks247

4.6.4 Construction of the Kneading Plane249

4.7 Piecewise Linear Circle Maps and Topological Entropy251

4.7.1 The Sawtooth Circle Map251

4.7.2 Circle Map with Given Kneading Sequences252

4.7.3 Kneading Determinant and Topological Entropy253

4.7.4 Construction of a Map from a Given Kneading Sequence255

4.7.5 Rotation Interval and Well-Ordered Periodic Sequences256

5 Symbolic Dynamics of Two-Dimensional Maps258

5.1 General Discussion260

5.1.1 Bi-Infinite Symbolic Sequences261

5.1.2 Decomposition of the Phase Plane263

5.1.3 Tangencies and Admissibility Conditions264

5.1.4 Admissibility Conditions in Symbolic Plane265

5.2 Invariant Manifolds and Dynamical Foliations of Phase Plane268

5.2.1 Stable and Unstable Invariant Manifolds268

5.2.2 Dynamical Foliations of the Phase Plane272

5.2.3 Summary and Discussion275

5.3 The Tél Map276

5.3.1 Forward and Backward Symbolic Sequences278

5.3.2 Dynamical Foliations of Phase Space and Their Ordering279

5.3.3 Forbidden and Allowed Zones in Symbolic Plane287

5.3.4 The Admissibility Conditions290

5.3.5 Summary294

5.4 The Lozi Map295

5.4.1 Forward and Backward Symbolic Sequences297

5.4.2 Dynamical Foliations of the Phase Space298

5.4.3 Ordering of the Forward and Backward Foliations307

5.4.4 Allowed and Forbidden Zones in the Symbolic Plane309

5.4.5 Discussion of the Admissibility Condition314

5.5 The Hénon Map315

5.5.1 Fixed Points and Their Stability317

5.5.2 Determination of Partition Lines in Phase Plane320

5.5.3 Hénon-Type Symbolic Dynamics325

5.5.4 Symbolic Analysis at Typical Parameter Values328

5.5.5 Discussion332

5.6 The Dissipative Standard Map338

5.6.1 Dynamical Foliations of the Phase Plane338

5.6.2 Ordering of Symbolic Sequences341

5.6.3 Symbolic Plane and Admissibility of Symbolic Sequences343

5.7 The Stadium Billiard Problem347

5.7.1 A Coding Based on Lifting349

5.7.2 Relation to Other Codings352

5.7.3 The Half-Stadium354

5.7.4 Summary356

6 Application to Ordinary Differential Equations357

6.1 General Discussion359

6.1.1 Three Types of ODEs359

6.1.2 On Numerical Integration of Differential Equations361

6.1.3 Numerical Calculation of the Poincaré Maps364

6.2 The Periodically Forced Brusselator368

6.2.1 The Brusselator Viewed from The Standard Map370

6.2.2 Transition from Annular to Interval Dynamics376

6.2.3 Symbolic Analysis of Interval Dynamics377

6.3 The Lorenz Equations385

6.3.1 Summary of Known Properties386

6.3.2 Construction of Poincaré and Return Maps390

6.3.3 One-Dimensional Symbolic Dynamics Analysis393

6.3.4 Symbolic Dynamics of the 2D Poincaré Maps396

6.3.5 Stable Periodic Orbits404

6.3.6 Concluding Remarks413

6.4 Summary of Other ODE Systems414

6.4.1 The Driven Two-Well Duffing Equation414

6.4.2 The NMR-Laser Model415

7 Counting the Number of Periodic Orbits417

7.1 Periodic versus Chaotic Regimes418

7.1.1 Stable Versus Unstable Periods in 1D Maps419

7.1.2 Notations and Summary of Results420

7.1.3 A Few Number Theory Notations and Functions424

7.2 Number of Periodic Orbits in a Class of One-Parameter Maps425

7.2.1 Number of Admissible Words in Symbolic Dynamics426

7.2.2 Number of Tangent and Period-Doubling Bifurcations427

7.2.3 Recursion Formula for the Total Number of Periods429

7.2.4 Symmetry Types of Periodic Sequences430

7.2.5 Explicit Solutions to the Recurrence Relations435

7.2.6 Finite Lambda Auto-Expansion of Real Numbers436

7.3 Other Aspects of the Counting Problem438

7.3.1 The Number of Roots of the“Dark Line”Equation438

7.3.2 Number of Saddle Nodes in Forming Smale Horseshoe439

7.3.3 Number of Solutions of Renormalization Group Equations439

7.4 Counting Formulae for General Continuous Maps441

7.5 Number of Periods in Maps With Discontinuity443

7.5.1 Number of Periods in the Gap Map444

7.5.2 Number of Periods in the Lorenz-Like Map446

7.6 Summary of the Counting Problem447

7.7 Cycle Expansion for Topological Entropy449

8 Symbolic Dynamics and Grammatical Complexity453

8.1 Formal Languages and Their Complexity455

8.1.1 Formal Language455

8.1.2 Chomsky Hierarchy of Grammatical Complexity456

8.1.3 The L-System458

8.2 Regular Language and Finite Automaton459

8.2.1 Finite Automaton459

8.2.2 Regular Language460

8.2.3 Stefan Matrix as Transfer Function for Automaton461

8.3 Beyond Regular Languages465

8.3.1 Feigenbaum and Generalized Feigenbaum Limiting Sets466

8.3.2 Even and Odd Fibonacci Sequences467

8.3.3 Odd Maximal Primitive Prefixes and Kneading Map469

8.3.4 Even Maximal Primitive Prefixes and Distinct Excluded Blocks472

8.4 Summary of Results473

9 Symbolic Dynamics and Knot Theory474

9.1 Knots and Links475

9.2 Knots and Links from Unimodal Maps477

9.3 Linking Numbers481

9.4 Discussion483

10 Appendix485

A.1 Program to Generate Admissible Sequences485

A.2 Program to Draw Dynamical Foliations of a Two-Dimensional Map491

A.3 A Greedy Program for Determining Partition Line496

References499

R.1 Books499

R.2 Papers500

Index517