Lai / Tél Transient Chaos

Complex Dynamics on Finite Time Scales
1. Auflage 2011
ISBN: 978-1-4419-6987-3
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 496 Seiten

The aim of this Book is to give an overview, based on the results of nearly three decades of intensive research, of transient chaos. One belief that motivates us to write this book is that, transient chaos may not have been appreciated even within the nonlinear-science community, let alone other scientific disciplines.

Autoren/Hrsg.

Lai, Ying-Cheng

Tél, Tamás

Inhaltsverzeichnis

1;Preface;6
2;Contents;12
3;Part I Basics of Transient Chaos;16
3.1;Chapter 1 Introduction to Transient Chaos;17
3.1.1;1.1 Basic Notions of Transient Chaos;20
3.1.1.1;1.1.1 Dynamical Systems;20
3.1.1.2;1.1.2 Saddles and Repellers;20
3.1.1.3;1.1.3 Types of Transient Chaos;23
3.1.2;1.2 Characterizing Transient Chaos;23
3.1.2.1;1.2.1 Escape Rate;24
3.1.2.2;1.2.2 Constructing Nonattracting Chaotic Sets;27
3.1.2.2.1;1.2.2.1 Horseshoe Construction;28
3.1.2.2.2;1.2.2.2 Ensemble Method;29
3.1.2.2.3;1.2.2.3 Sprinkler Method;31
3.1.2.2.4;1.2.2.4 Single-Trajectory (PIM-Triple) Method;31
3.1.2.3;1.2.3 The Invariant Measures of Transient Chaos;32
3.1.2.3.1;1.2.3.1 Natural Measure;32
3.1.2.3.2;1.2.3.2 Conditionally Invariant Measure;33
3.1.2.3.3;1.2.3.3 Characterization of the Natural Measure;36
3.1.3;1.3 Experimental Evidence of Transient Chaos;39
3.1.3.1;1.3.1 Convection Loop Experiment;39
3.1.3.2;1.3.2 Chemical Reactions Preceding Thermal Equilibrium;40
3.1.3.3;1.3.3 Nuclear Magnetic Resonance Laser Experiment;40
3.1.3.4;1.3.4 Driven Pendulum;41
3.1.3.5;1.3.5 Fractal Basin Boundaries;42
3.1.3.6;1.3.6 Advection in the Wake of a Cylinder;43
3.1.3.7;1.3.7 Semiclassical Fluctuations in Chaotic Scattering;44
3.1.3.8;1.3.8 Emission of Light from Dielectric Cavities;45
3.1.3.9;1.3.9 Maintaining Chaos in a Magnetoelastic Ribbon;46
3.1.3.10;1.3.10 Turbulence in Pipe Flows;46
3.1.4;1.4 A Brief History of Transient Chaos;48
3.2;Chapter 2 Transient Chaos in Low-Dimensional Systems;50
3.2.1;2.1 One-Dimensional Maps, Natural Measures, and c-Measures;51
3.2.1.1;2.1.1 Basic Properties of One-Dimensional Maps Generating Transient Chaos;51
3.2.1.2;2.1.2 Conditionally Invariant Measure;53
3.2.1.3;2.1.3 The Frobenius–Perron Equation;54
3.2.2;2.2 General Relations;57
3.2.2.1;2.2.1 Lyapunov Exponent, Information Dimension, and Metric Entropy;57
3.2.2.2;2.2.2 Box-Counting Dimension and Topological Entropy;58
3.2.2.3;2.2.3 An Analytically Tractable Example: The Tent Map;60
3.2.3;2.3 Examples of Transient Chaos in One Dimension;61
3.2.3.1;2.3.1 Numbers with Incomplete Continued Fractions;61
3.2.3.2;2.3.2 Shimmying Wheels;64
3.2.3.3;2.3.3 Random-Field Ising Chain;66
3.2.4;2.4 Nonhyperbolic Transient Chaos in One Dimension and Intermittency;68
3.2.5;2.5 Analytic Example of Transient Chaos in Two Dimensions;71
3.2.6;2.6 General Properties of Chaotic Saddlesin Two-Dimensional Maps;75
3.2.6.1;2.6.1 Natural Measure and c-Measure;75
3.2.6.2;2.6.2 Entropy and Dimension Formulas;77
3.2.6.3;2.6.3 Information-Theoretic Arguments;79
3.2.6.4;2.6.4 Organization About Unstable Periodic Orbits;80
3.2.7;2.7 Leaked Dynamical Systems and Poincaré Recurrences;83
3.2.7.1;2.7.1 Chaotic Saddles Associated with Leaked Systems;83
3.2.7.2;2.7.2 Poincaré Recurrences;87
3.3;Chapter 3 Crises;91
3.3.1;3.1 Boundary Crises;92
3.3.1.1;3.1.1 Nonhyperbolicity of Chaotic Saddles;95
3.3.1.2;3.1.2 Critical Exponent of Chaotic Transients;98
3.3.1.2.1;3.1.2.1 Heteroclinic Tangency;98
3.3.1.2.2;3.1.2.2 Homoclinic Tangency;100
3.3.1.2.3;3.1.2.3 One-Dimensional Maps;102
3.3.2;3.2 Interior Crises;102
3.3.2.1;3.2.1 An Example of Interior Crisis;103
3.3.2.2;3.2.2 Periodic Windows;105
3.3.3;3.3 Crisis-Induced Intermittency;110
3.3.3.1;3.3.1 Example of Basic Components: One-Dimensional Map;112
3.3.3.2;3.3.2 Example of Basic Components: Two-Dimensional Map;114
3.3.4;3.4 Gap-Filling and Growth of Topological Entropy;115
3.4;Chapter 4 Noise and Transient Chaos;119
3.4.1;4.1 Effects of Noise on Lifetime of Transient Chaos;120
3.4.1.1;4.1.1 General Setting;120
3.4.1.2;4.1.2 Enhancement of Transient Lifetime by Noise;121
3.4.2;4.2 Quasipotentials;123
3.4.2.1;4.2.1 Basic Notions;123
3.4.2.2;4.2.2 Quasipotential Plateaus Associated with Nonattracting Chaotic Sets;126
3.4.2.3;4.2.3 Exit Rates from Attractor and Most Probable Exit Paths;128
3.4.2.4;4.2.4 Enhancement of Exit Rates by Transient Chaos;129
3.4.3;4.3 Noise-Induced Chaos;131
3.4.3.1;4.3.1 Critical Noise Strength for Noise-Induced Chaos;132
3.4.3.2;4.3.2 Scaling Laws for Critical Noise Strength and for Lifetime at a Saddle-Node Bifurcation;134
3.4.3.3;4.3.3 Appearance of a Positive Lyapunov Exponent;135
3.4.3.4;4.3.4 Scaling Law for the Largest Lyapunov Exponent;136
3.4.4;4.4 General Properties of Noise-Induced Chaos;140
3.4.4.1;4.4.1 Fractal Properties;140
3.4.4.2;4.4.2 Noise-Induced Unstable Dimension Variability;141
3.4.4.3;4.4.3 Ubiquitous Applications to Biological Sciences;143
3.4.5;4.5 Noise-Induced Crisis;144
3.4.6;4.6 Random Maps and Transient Phenomena;146
3.4.6.1;4.6.1 Open Random Maps, Snapshot Chaotic Saddles;148
3.4.6.2;4.6.2 Transient Behavior in Fractal Snapshot Attractors;151
4;Part II Physical Manifestations of Transient Chaos;156
4.1;Chapter 5 Fractal Basin Boundaries;157
4.1.1;5.1 Basin Boundaries: Basics;158
4.1.2;5.2 Types of Fractal Basin Boundaries;159
4.1.2.1;5.2.1 Filamentary Fractal Boundaries;160
4.1.2.2;5.2.2 Continuous Fractal Boundaries;160
4.1.2.3;5.2.3 Sporadically Fractal Boundaries;161
4.1.2.4;5.2.4 Riddled Basins;162
4.1.3;5.3 Fractal Basin Boundaries and Predictability;163
4.1.4;5.4 Emergence of Fractal Basin Boundaries;168
4.1.4.1;5.4.1 Basin Boundary Metamorphoses and Accessible Orbits;168
4.1.4.2;5.4.2 Dimension Changes at Basin Boundary Metamorphoses;170
4.1.4.3;5.4.3 A Two-Dimensional Model;173
4.1.5;5.5 Wada Basin Boundaries;175
4.1.6;5.6 Sporadically Fractal Basin Boundaries;180
4.1.6.1;5.6.1 Chaotic Phase Synchronization;180
4.1.6.2;5.6.2 Dynamical Mechanism;182
4.1.7;5.7 Riddled Basins;185
4.1.7.1;5.7.1 Riddling Bifurcation;186
4.1.7.2;5.7.2 An Example;187
4.1.7.3;5.7.3 Scaling Relation;188
4.1.8;5.8 Catastrophic Bifurcation of a Riddled Basin;189
4.1.8.1;5.8.1 An Example;189
4.1.8.2;5.8.2 Critical Behavior and Scaling Laws;192
4.2;Chapter 6 Chaotic Scattering;196
4.2.1;6.1 Occurrence of Scattering;197
4.2.2;6.2 A Paradigmatic Example of Chaotic Scattering;199
4.2.3;6.3 Transitions to Chaotic Scattering;204
4.2.3.1;6.3.1 Scattering from a Single Hill;205
4.2.3.2;6.3.2 Abrupt Bifurcation to Chaotic Scattering;206
4.2.3.2.1;6.3.2.1 Basic Phenomenon;206
4.2.3.2.2;6.3.2.2 Scaling of Dynamical Invariants with Energy;209
4.2.3.3;6.3.3 Saddle-Center Bifurcation to Chaotic Scattering;212
4.2.3.4;6.3.4 Abrupt Bifurcation to Chaotic Scattering with Discontinuous Change in Dimension;216
4.2.4;6.4 Nonhyperbolic Chaotic Scattering;220
4.2.4.1;6.4.1 Algebraic Decay;220
4.2.4.2;6.4.2 Development of Horseshoe Structure in Nonhyperbolic Chaotic Scattering;222
4.2.4.3;6.4.3 Dimension in Nonhyperbolic Chaotic Scattering;225
4.2.4.4;6.4.4 Intermediate-Time Exponential Decay;227
4.2.4.5;6.4.5 Relation to Poincaré Recurrences;228
4.2.5;6.5 Fluctuations of the Algebraic-Decay Exponent in Nonhyperbolic Chaotic Scattering;231
4.2.5.1;6.5.1 Numerical Model;232
4.2.5.2;6.5.2 Decay-Exponent Fluctuations;234
4.2.6;6.6 Effect of Dissipation and Noise on Chaotic Scattering;239
4.2.7;6.7 Application of Nonhyperbolic Chaotic Scattering: Dynamics in Deformed Optical Microlasing Cavities;241
4.2.7.1;6.7.1 Dynamical Criterion for High-Q Operation;243
4.2.7.2;6.7.2 A Numerical Example;244
4.3;Chapter 7 Quantum Chaotic Scattering and Conductance Fluctuations in Nanostructures;248
4.3.1;7.1 Quantum Manifestation of Chaotic Scattering;249
4.3.2;7.2 Hyperbolic Chaotic Scattering;251
4.3.2.1;7.2.1 Autocorrelation of the S-Matrix Elements;251
4.3.2.2;7.2.2 S-Matrix in the Time Domain;251
4.3.2.3;7.2.3 Relation to Orthogonal Ensemble of Random Matrices;252
4.3.3;7.3 Nonhyperbolic Chaotic Scattering;254
4.3.4;7.4 Conductance Fluctuations in Quantum Dots;256
4.3.4.1;7.4.1 Basic Physics of Quantum Dots;257
4.3.4.2;7.4.2 Büttiker–Landauer Formula;259
4.3.4.3;7.4.3 Conductance Fluctuations as Quantum Manifestation of Chaotic Scattering;261
4.3.5;7.5 Dynamical Tunneling in Nonhyperbolic Quantum Dots;263
4.3.6;7.6 Dynamical Tunneling and Quantum Echoes in Scattering;268
4.3.7;7.7 Leaked Quantum Systems;270
5;Part III High-Dimensional Transient Chaos;272
5.1;Chapter 8 Transient Chaos in Higher Dimensions;273
5.1.1;8.1 Three-Dimensional Open Baker Map;274
5.1.2;8.2 Escape Rate, Entropies, and Fractal Dimensionsfor Nonattracting Chaotic Sets in Higher Dimensions;276
5.1.2.1;8.2.1 Escape Rate and Entropies;276
5.1.2.2;8.2.2 Dimension Formulas for High-Dimensional Chaotic Saddles;278
5.1.3;8.3 Models Testing Dimension Formulas;282
5.1.3.1;8.3.1 Two-Dimensional Noninvertible Map Model;282
5.1.3.1.1;8.3.1.1 Natural Measure and Lyapunov Exponents;282
5.1.3.1.2;8.3.1.2 Dimension Formulas;284
5.1.3.1.3;8.3.1.3 The Issue of Typicality;286
5.1.3.2;8.3.2 A Chaotic Billiard Scatterer;287
5.1.4;8.4 Numerical Method for Computing High-Dimensional Chaotic Saddles: Stagger-and-Step;290
5.1.4.1;8.4.1 Basic Idea;290
5.1.4.2;8.4.2 Invariant Sets Constrained to Slow Manifolds;292
5.1.5;8.5 High-Dimensional Chaotic Scattering;295
5.1.5.1;8.5.1 Dimension Requirement for Chaotic Saddles to be Observables;295
5.1.5.2;8.5.2 Normally Hyperbolic Invariant Manifolds in High-Dimensional Chaotic Scattering;297
5.1.5.3;8.5.3 Metamorphosis in High-Dimensional Chaotic Scattering;299
5.1.5.4;8.5.4 Topological Change Accompanying the Metamorphosis;304
5.1.6;8.6 Superpersistent Transient Chaos: Basics;306
5.1.6.1;8.6.1 Unstable–Unstable Pair Bifurcation;306
5.1.6.2;8.6.2 Riddling Bifurcation and Superpersistent Chaotic Transients;309
5.1.7;8.7 Superpersistent Transient Chaos: Effect of Noiseand Applications;313
5.1.7.1;8.7.1 Noise-Induced Superpersistent Chaotic Transients;313
5.1.7.2;8.7.2 Application: Advection of Inertial Particles in Open Chaotic Flows;316
5.2;Chapter 9 Transient Chaos in Spatially Extended Systems;319
5.2.1;9.1 Basic Characteristics of Spatiotemporal Chaos;320
5.2.1.1;9.1.1 Paradigmatic Models;320
5.2.1.2;9.1.2 Phase Spaces of Spatiotemporal Systems;321
5.2.1.3;9.1.3 Spatiotemporal Intermittency;323
5.2.2;9.2 Supertransients;324
5.2.2.1;9.2.1 Transient Chaos in Coupled Map Lattices;324
5.2.2.2;9.2.2 Origin of Supertransient Scaling;325
5.2.2.3;9.2.3 Supertransients with Exponentially Long Lifetimes in Other Systems;327
5.2.2.4;9.2.4 Stable Chaos;328
5.2.3;9.3 Effect of Noise and Nonlocal Coupling on Supertransients;329
5.2.4;9.4 Crises in Spatiotemporal Dynamical Systems;331
5.2.4.1;9.4.1 Boundary Crises: Supertransients Preceding Asymptotic Spatiotemporal Chaos;331
5.2.4.2;9.4.2 Interior Crises in Spatially Coherent Chaotic Systems;332
5.2.4.3;9.4.3 Crises Leading to Fully Developed Spatiotemporal Chaos;335
5.2.5;9.5 Fractal Properties of Supertransients;337
5.2.5.1;9.5.1 Dimensions;337
5.2.5.2;9.5.2 Dimension Densities;340
5.2.6;9.6 Turbulence in Pipe Flows;341
5.2.6.1;9.6.1 Turbulence Lifetime;341
5.2.6.2;9.6.2 Other Aspects of Hydrodynamical Supertransients;345
5.2.7;9.7 Closing Remarks;346
6;Part IV Applications of Transient Chaos;348
6.1;Chapter 10 Chaotic Advection in Fluid Flows;349
6.1.1;10.1 General Setting of Passive Advective Dynamics;350
6.1.2;10.2 Passive Advection in von Kármán Vortex Streets;352
6.1.2.1;10.2.1 Flow Model;352
6.1.2.2;10.2.2 Advection and Droplet Dynamics;354
6.1.3;10.3 Point Vortex Problems;357
6.1.3.1;10.3.1 Vortex Dynamics;357
6.1.3.2;10.3.2 Advection by Leapfrogging Vortex Pairs;359
6.1.3.3;10.3.3 Lobe Dynamics;362
6.1.4;10.4 Dye Boundaries;364
6.1.5;10.5 Advection in Aperiodic Flows;368
6.1.5.1;10.5.1 Coherent Structures in Aperiodic Flows;369
6.1.5.2;10.5.2 Open Aperiodic Flows;372
6.1.6;10.6 Advection in Closed Flows with Leaks;376
6.1.7;10.7 Advection of Finite-Size Particles;379
6.1.8;10.8 Reactions in Open Flows;383
6.1.8.1;10.8.1 Heuristic Theory;384
6.1.8.2;10.8.2 Biological Activities;387
6.1.8.3;10.8.3 Reactions in Open Aperiodic Flows;387
6.2;Chapter 11 Controlling Transient Chaos and Applications;390
6.2.1;11.1 Controlling Transient Chaos: General Introduction;391
6.2.1.1;11.1.1 Basic Idea and Method;391
6.2.1.2;11.1.2 Scaling Laws Associated with Control;393
6.2.1.3;11.1.3 Remarks;396
6.2.1.3.1;11.1.3.1 Controlling Fractal Basin Boundaries;396
6.2.1.3.2;11.1.3.2 Controlling Chaotic Scattering;396
6.2.1.3.3;11.1.3.3 Improved Method of Controlling a Chaotic Saddle;396
6.2.2;11.2 Maintaining Chaos: General Introduction;397
6.2.2.1;11.2.1 Basic Idea;397
6.2.2.2;11.2.2 Maintaining Chaos Using a Periodic Saddle Orbit;398
6.2.2.3;11.2.3 Practical Method of Control;399
6.2.3;11.3 Voltage Collapse and Prevention;400
6.2.3.1;11.3.1 Modeling Voltage Collapse in Electrical Power Systems;400
6.2.3.2;11.3.2 Example of Control;402
6.2.4;11.4 Maintaining Chaos to Prevent Species Extinction;404
6.2.4.1;11.4.1 Food-Chain Model;405
6.2.4.2;11.4.2 Dynamical Mechanism of Species Extinction;406
6.2.4.3;11.4.3 Control to Prevent Species Extinction;406
6.2.5;11.5 Maintaining Chaos in the Presence of Noise, Safe Sets;410
6.2.6;11.6 Encoding Digital Information Using Transient Chaos;412
6.2.6.1;11.6.1 The Channel Capacity;413
6.2.6.2;11.6.2 Message Encoding, Control Scheme, and Noise Immunity;413
6.3;Chapter 12 Transient Chaotic Time-Series Analysis;418
6.3.1;12.1 Reconstruction of Phase Space;419
6.3.1.1;12.1.1 Reconstruction of Invariant Sets;421
6.3.1.2;12.1.2 Reconstructing Invariant Sets of Delay-Differential Equations;424
6.3.2;12.2 Detection of Unstable Periodic Orbits;426
6.3.2.1;12.2.1 Extracting Unstable Periodic Orbits from Transient Chaotic Time Series;426
6.3.2.2;12.2.2 Detectability of Unstable Periodic Orbits from Transient Chaotic Time Series;429
6.3.3;12.3 Computation of Dimension;431
6.3.3.1;12.3.1 Basics;431
6.3.3.2;12.3.2 Applicability to Transient Chaotic Time Series;433
6.3.4;12.4 Computing Lyapunov Exponents from Transient Chaotic Time Series;435
6.3.4.1;12.4.1 Searching for Neighbors in the Embedding Space;435
6.3.4.2;12.4.2 Computing the Tangent Maps;436
6.3.4.3;12.4.3 Computing the Exponents;437
6.3.4.4;12.4.4 A Numerical Example;438
6.3.4.5;12.4.5 Remarks;438
7;Final Remarks;440
8;Appendix A Multifractal Spectra;441
8.1;A.1 Definition of Spectra;441
8.2;A.2 Multifractal Spectra for Repellers of One-Dimensional Maps;441
8.3;A.3 Multifractal Spectra of Saddles of Two-Dimensional Maps;445
8.4;A.4 Zeta Functions;446
9;Appendix B Open Random Baker Maps;448
9.1;B.1 Single Scale Baker Map;448
9.2;B.2 General Baker Map;450
10;Appendix C Semiclassical Approximation;452
10.1;C.1 Semiclassical S-Matrix in Action-Angle Representation;452
10.2;C.2 Stationary Phase Approximation and the Maslov Index;453
11;Appendix D Scattering Cross Sections;457
11.1;D.1 Scattering Cross Sections in Classical Chaotic Scattering;457
11.2;D.2 Semiclassical Scattering Cross Sections;459
12;References;461
13;Index;493