Transient Chaos (eBook)

Ying-Cheng Lai’s main research interests are Chaotic Dynamics, Complex Networks, Quantum Transport in Nanostructures, Biological Physics, and Signal Processing. He is a professor at the Arizona State University. Tamas Tel's main research interests are Chaotic Dynamics, Stochastic Processes, Hydrodynamics, Spreading of Pollutants, and Environmental Flows.

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