Elementary symbolic dynamics and chaos in dissipative systems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore u.a.
World Scientific
1989
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 460 S. |
| ISBN: | 997150698X |
Staff View
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Record in the Search Index
| _version_ | 1804117928121991168 |
|---|---|
| adam_text | Contents
Preface v
1 Mathematical Models Exhibiting Chaos 1
1.1 What is Chaos? 1
1.2 The Concept of Universality and the Role of Models 4
1.3 Insect Population and the Logistic Map 6
1.4 Thermal Convection, the Lorenz Model and the
Antisymmetric Cubic Map 11
1.5 Chemical Kinetics Models and Forced Limit
Cycle Oscillators 16
1.6 Optical Bistability and the Sine Square Map 20
2 One Dimensional Mappings 25
2.1 The Skeleton of Bifurcation Diagrams 25
2.1.1 The Logistic Map 27
2.1.2 The Sine Square Map 30
2.1.3 The Rainbow: Explanation of the Dark Lines 31
2.2 Mathematical Preliminaries 36
2.2.1 Composite Functions 36
2.2.2 The Chain Rule of Differentiation 36
2.2.3 The Implicit Function Theorem 37
2.2.4 The Schwarzian Derivative of a Function 38
2.3 Stable and Superstable Orbits 40
2.3.1 Fixed Point and Its Stability 43
ix
x Contents
2.3.2 Periodic Orbits and Their Stability 44
2.4 The Period doubling Cascade 46
2.5 Chaotic Nature of the Surjective Logistic Map 50
2.5.1 Principal Branch of the Arccosine Function 51
2.5.2 The Surjective Tent Map 51
2.5.3 Topological Conjugacy of Maps 53
2.5.4 The Invariant Distribution for the Surjective
Logistic Map 55
2.5.5 Homoclinic Points in the Logistic Map 57
2.6 General Analysis of Tangent and Period doubling
Bifurcations 60
2.6.1 The Period 3 Window 60
2.6.2 Tangent Bifurcation 62
2.6.3 Intermittency 66
2.6.4 Period doubling Bifurcation 70
2.7 The Renormalization Group Equation for Period doubling ... 75
2.7.1 Renormalization Group Idea in Phase Transition
Theory 76
2.7.2 Derivation of the Feigenbaum Renormalization
Group Equation 79
2.7.3 Linearized Renormalization Group Equation
and Convergence Rate S 85
2.7.4 A Generalized Renormalization Group Equation .... 87
2.8 Scaling Properties Related to External Noise 88
2.9 Renormalization Group Equation for Intermittency 91
2.9.1 The Scaling Theory and Generalized
Homogeneous Functions 91
2.9.2 Scaling Theory for Intermittency 94
2.9.3 Renormalization Group Equation and Its
Exact Solution 96
2.9.4 Linearized Renormalization Group
Equations and the Exponents 99
Contents xi
3 Elementary Symbolic Dynamics 101
3.1 Introduction 102
3.2 The Location of Superstable Orbits 104
3.2.1 Word Lifting Technique 105
3.2.2 Example 1: The Logistic Map 106
3.2.3 Example 2: The Sine square Map 108
3.2.4 Generalizations Ill
3.3 Symbolic Dynamics of Two Letters 112
3.3.1 The Ordering of Words 113
3.3.2 Finding the Median Word Between
Two Given Words 115
3.3.3 A Program to Generate Median Words 119
3.3.4 The * composition and Fine Structure of
Power Spectra 122
3.4 Scaling Property of the Period n tupling Sequences 125
3.4.1 The Convergence Rate Sw 125
3.4.2 The Renormalization Group Equations 126
3.4.3 Estimating the Scaling Factors a^ 130
3.4.4 Numerical Solution of the Renormalization
Group Equations 131
3.5 Admissibility of Words and the Tent Map 133
3.5.1 The Tent Map 134
3.5.2 A expansions of Real Numbers 136
3.5.3 Method to Generate All Admissible Words
of Given Length 139
3.6 Symbolic Dynamics of Two Letters Revisited 140
3.6.1 Monotonicity of Functions and Ordering of Words . . . 141
3.6.2 Auxiliary Notation 142
3.6.3 Maximal Sequences and Admissibility of Words 144
3.6.4 The Periodic Window Theorem 146
3.6.5 The Median Sequence In Between Two
Given Sequences 149
3.6.6 The Generalized Composition Rule 151
xii Contents
3.6.7 Applications of the Generalized
Composition Rule 152
3.7 Symbolic Dynamics of Three and More Letters 160
3.7.1 The Antisymmetric Cubic Map 161
3.7.2 The Ordering of Sequences 165
3.7.3 Construction of Median Sequences 168
3.7.4 Symbolic Dynamics of Four Letters 174
3.8 Symmetry Breaking and Restoration in Antisymmetric
mappings 178
3.8.1 Symmetry Breaking Bifurcation 180
3.8.2 Symbolic Dynamics Analysis of Symmetry Breakings . . 185
3.8.3 Symbolic Dynamics and Symmetry Restoration 187
3.9 The Number of Periodic Windows 191
3.9.1 A Recursion Formula for the Number of Periodic
Windows 192
3.9.2 The Necklace Problem — Symmetry Types of
Periodic Sequences 196
3.9.3 The Number of Saddle Orbits in the Horseshoe 197
3.9.4 Number of Primitive Admissible Words 200
3.9.5 Number of Periods in the Antisymmetric Cubic Map . . 201
4 Circle Mappings and Two Dimensional Maps 205
4.1 The Physics of Linear and Nonlinear Oscillators 206
4.2 The Bare Circle Map 208
4.3 General Circle Maps 212
4.4 Farey Tree and Fibonacci Numbers 216
4.4.1 Farey Tree: Rational Fraction Representation 216
4.4.2 Farey Tree: Continued Fraction Representation 217
4.4.3 Farey Tree: Symbolic Representation 220
4.4.4 The W sequence: Ordering of Rotation Numbers .... 224
4.4.5 The Golden Mean and Fibonacci Numbers 228
4.5 How the Arnold Tongues Become
Sausages: a Piecewise Linear Map 229
4.5.1 The Critical Line 230
Contents xiii
4.5.2 Structure of the Mode Locking Tongues 232
4.5.3 Shrinking Points of the Arnold Tongues 235
4.6 The He non Map 239
4.6.1 Fixed Points and Their Stability 241
4.6.2 Stable Periodic Orbits for the Henon Map 243
4.6.3 Stable and Unstable Invariant Manifolds 246
4.7 The Dissipative Standard Map 250
5 Chaos in Ordinary Differential Equations 253
5.1 Three Kinds of Ordinary Differential Equations 254
5.2 On Numerical Integration of Differential Equations 256
5.3 Numerical Calculation of the Poincare Maps 259
5.3.1 Autonomous Systems: He non s Method 260
5.3.2 Non autonomous Systems: Subharmonic
Stroboscopic Sampling 262
5.3.3 An Atlas of Maps 265
5.3.4 Rotation Numbers and Symbolic Dynamics 269
5.4 Technique for Location of Periodic Orbits 274
5.4.1 Non autonomous Systems 275
5.4.2 Autonomous Systems 276
5.4.3 Summary of Periodic Orbit Locating Technique 279
5.5 Visualization of the Dynamics 279
5.6 Power Spectrum Analysis 282
5.6.1 Preliminaries 283
5.6.2 Design of the Spectrum 284
5.6.3 Symbolic Dynamics and Fine Structure of Spectra . . . 285
5.7 Case Study 1: the Forced Brusselator 287
5.7.1 Linear Stability Analysis 289
5.7.2 The Existence of Periodic Solutions 291
5.7.3 Hierarchy of Chaotic Bands and the U sequence .... 294
5.7.4 Transitions from Quasiperiodicity to Chaos 303
5.7.5 Homoclinic and Heteroclinic Intersections 311
5.8 Case Study 2: the Lorenz model 326
5.8.1 Nomenclature of Periods for the Lorenz Model 326
xiv Contents
5.8.2 Systematics of Periods in the Lorenz Model 329
6 Characterization of Chaotic Attractors 337
6.1 Various Definitions of Dimensions 338
6.1.1 The Fractal Dimension Do 339
6.1.2 The Information Dimension D 342
6.1.3 The Correlation Dimension D2 343
6.1.4 High Order Information Dimensions Dq 347
6.1.5 Derivation of the Sum Rule (6.11) 350
6.1.6 Dq for the Limiting Sets of Period n tupling
Sequences 354
6.1.7 Calculation of Dq from Time Series 360
6.2 Thermodynamic Formalism for Multifractals 362
6.2.1 The Singularity Point of View 362
6.2.2 Phase Transitions in the Thermodynamic Formalism . . 368
6.3 Time difference Method for the Reconstruction of
Phase Space Dynamics 375
6.3.1 On the Choice of the Embedding Dimension 376
6.3.2 On the Choice of the Time Difference 377
6.4 The Lyapunov Exponents 379
6.4.1 Invoking the Tangent Space — the Key to Success . . . 381
6.4.2 General Considerations 383
6.4.3 Calculation of Lyapunov Exponents from the
Evolution Equations 385
6.4.4 Lyapunov Exponents for Periodic Orbits 389
6.4.5 The Most Stable Manifold and Destruction of
Invariant Tori 389
6.4.6 Calculation of Lyapunov Exponents from
Experimental Data 396
6.5 Information and Entropy 403
6.5.1 The Metric Entropy 404
6.5.2 The Topological Entropy 405
6.6 Relation Between Dimension, Lyapunov Exponents
and Entropy 411
Contents xv
6.7 Peculiarity of One dimensional Mappings 412
7 Transient Behaviour 417
7.1 Critical Slowing Down Near Period doubling Bifurcations . . . 420
7.2 Transient Precursor of Bifurcations 422
7.3 Chaotic Transients 425
7.4 Escape Rate from Strange Repellers 427
References 429
Subject Index 457
|
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| indexdate | 2024-07-09T16:02:11Z |
| institution | BVB |
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| language | English |
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| spelling | Hao, Bailin Verfasser aut Elementary symbolic dynamics and chaos in dissipative systems Hao Bai-lin Singapore u.a. World Scientific 1989 XV, 460 S. txt rdacontent n rdamedia nc rdacarrier Chaos Chaos gtt Comportement chaotique des systèmes ram Differentieerbaarheid gtt Dynamique différentiable Dynamische systemen gtt Systèmes dynamiques différentiables ram chaos inriac comportement chaotique inriac système différentiable inriac système dynamique inriac Chaotic behavior in systems Symbolic dynamics Dissipatives System (DE-588)4209641-8 gnd rswk-swf Dynamik (DE-588)4013384-9 gnd rswk-swf Chaos (DE-588)4191419-3 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Bifurkation Geografie (DE-588)4145371-2 gnd rswk-swf Dissipatives System (DE-588)4209641-8 s Chaos (DE-588)4191419-3 s DE-604 Bifurkation Geografie (DE-588)4145371-2 s 1\p DE-604 Dynamik (DE-588)4013384-9 s 2\p DE-604 Mathematisches Modell (DE-588)4114528-8 s 3\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002277965&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Hao, Bailin Elementary symbolic dynamics and chaos in dissipative systems Chaos Chaos gtt Comportement chaotique des systèmes ram Differentieerbaarheid gtt Dynamique différentiable Dynamische systemen gtt Systèmes dynamiques différentiables ram chaos inriac comportement chaotique inriac système différentiable inriac système dynamique inriac Chaotic behavior in systems Symbolic dynamics Dissipatives System (DE-588)4209641-8 gnd Dynamik (DE-588)4013384-9 gnd Chaos (DE-588)4191419-3 gnd Mathematisches Modell (DE-588)4114528-8 gnd Bifurkation Geografie (DE-588)4145371-2 gnd |
| subject_GND | (DE-588)4209641-8 (DE-588)4013384-9 (DE-588)4191419-3 (DE-588)4114528-8 (DE-588)4145371-2 |
| title | Elementary symbolic dynamics and chaos in dissipative systems |
| title_auth | Elementary symbolic dynamics and chaos in dissipative systems |
| title_exact_search | Elementary symbolic dynamics and chaos in dissipative systems |
| title_full | Elementary symbolic dynamics and chaos in dissipative systems Hao Bai-lin |
| title_fullStr | Elementary symbolic dynamics and chaos in dissipative systems Hao Bai-lin |
| title_full_unstemmed | Elementary symbolic dynamics and chaos in dissipative systems Hao Bai-lin |
| title_short | Elementary symbolic dynamics and chaos in dissipative systems |
| title_sort | elementary symbolic dynamics and chaos in dissipative systems |
| topic | Chaos Chaos gtt Comportement chaotique des systèmes ram Differentieerbaarheid gtt Dynamique différentiable Dynamische systemen gtt Systèmes dynamiques différentiables ram chaos inriac comportement chaotique inriac système différentiable inriac système dynamique inriac Chaotic behavior in systems Symbolic dynamics Dissipatives System (DE-588)4209641-8 gnd Dynamik (DE-588)4013384-9 gnd Chaos (DE-588)4191419-3 gnd Mathematisches Modell (DE-588)4114528-8 gnd Bifurkation Geografie (DE-588)4145371-2 gnd |
| topic_facet | Chaos Comportement chaotique des systèmes Differentieerbaarheid Dynamique différentiable Dynamische systemen Systèmes dynamiques différentiables chaos comportement chaotique système différentiable système dynamique Chaotic behavior in systems Symbolic dynamics Dissipatives System Dynamik Mathematisches Modell Bifurkation Geografie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002277965&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT haobailin elementarysymbolicdynamicsandchaosindissipativesystems |