Part I Basics of Transient Chaos ............................... 1
1 Introduction to Transient Chaos .............................. 3
1.1 Basic Notions of Transient Chaos ........................ 6
1.2 Characterizing Transient Chaos .......................... 9
1.3 Experimental Evidence of Transient Chaos ............... 25
1.4 A Brief History of Transient Chaos ..................... 34
2 Transient Chaos in Low-Dimensional Systems .................. 37
2.1 One-Dimensional Maps, Natural Measures, and
c-Measures ............................................. 38
2.2 General Relations ...................................... 44
2.3 Examples of Transient Chaos in One Dimension ........... 48
2.4 Nonhyperbolic Transient Chaos in One Dimension
and Intermittency ...................................... 55
2.5 Analytic Example of Transient Chaos in Two
Dimensions ............................................. 58
2.6 General Properties of Chaotic Saddles in Two-
Dimensional Maps ....................................... 62
2.7 Leaked Dynamical Systems and Poincare Recurrences ...... 70
3 Crises ...................................................... 79
3.1 Boundary Crises ........................................ 80
3.2 Interior Crises ........................................ 90
3.3 Crisis-Induced Intermittency ........................... 98
3.4 Gap-Filling and Growth of Topological Entropy ......... 103
4 Noise and Transient Chaos .................................. 107
4.1 Effects of Noise on Lifetime of Transient Chaos ....... 108
4.2 Quasipotentials ....................................... 111
4.3 Noise-Induced Chaos ................................... 119
4.4 General Properties of Noise-Induced Chaos ............. 128
4.5 Noise-Induced Crisis .................................. 132
4.6 Random Maps and Transient Phenomena ................... 134
Part II Physical Manifestations of Transient Chaos
5 Fractal Basin Boundaries ................................... 147
5.1 Basin Boundaries: Basics .............................. 148
5.2 Types of Fractal Basin Boundaries ..................... 149
5.3 Fractal Basin Boundaries and Predictability ........... 153
5.4 Emergence of Fractal Basin Boundaries ................. 158
5.5 Wada Basin Boundaries ................................. 165
5.6 Sporadically Fractal Basin Boundaries ................. 170
5.7 Riddled Basins ........................................ 175
5.8 Catastrophic Bifurcation of a Riddled Basin ........... 179
6 Chaotic Scattering ......................................... 187
6.1 Occurrence of Scattering .............................. 188
6.2 A Paradigmatic Example of Chaotic Scattering .......... 190
6.3 Transitions to Chaotic Scattering ..................... 195
6.4 Nonhyperbolic Chaotic Scattering ...................... 211
6.5 Fluctuations of the Algebraic-Decay Exponent
in Nonhyperbolic Chaotic Scattering ................... 222
6.6 Effect of Dissipation and Noise on Chaotic
Scattering ............................................ 230
6.7 Application of Nonhyperbolic Chaotic Scattering:
Dynamics in Deformed Optical Microlasing Cavities ..... 232
7 Quantum Chaotic Scattering and Conductance Fluctuations
in Nanostructures .......................................... 239
7.1 Quantum Manifestation of Chaotic Scattering ........... 240
7.2 Hyperbolic Chaotic Scattering ......................... 242
7.3 Nonhyperbolic Chaotic Scattering ...................... 245
7.4 Conductance Fluctuations in Quantum Dots .............. 247
7.5 Dynamical Tunneling in Nonhyperbolic Quantum Dots ..... 254
7.6 Dynamical Tunneling and Quantum Echoes in Scattering .. 259
7.7 Leaked Quantum Systems ................................ 261
Part III High-Dimensional Transient Chaos
8 Transient Chaos in Higher Dimensions ....................... 265
8.1 Three-Dimensional Open Baker Map ...................... 266
8.2 Escape Rate, Entropies, and Fractal Dimensions for
Nonattracting Chaotic Sets in Higher Dimensions ....... 268
8.3 Models Testing Dimension Formulas ..................... 274
8.4 Numerical Method for Computing High-Dimensional
Chaotic Saddles: Stagger-and-Step ..................... 282
8.5 High-Dimensional Chaotic Scattering ................... 287
8.6 Superpersistent Transient Chaos: Basics ............... 298
8.7 Superpersistent Transient Chaos: Effect of Noise
and Applications ...................................... 305
9 Transient Chaos in Spatially Extended Systems .............. 311
9.1 Basic Characteristics of Spatiotemporal Chaos ......... 312
9.2 Supertransients ....................................... 316
9.3 Effect of Noise and Nonlocal Coupling on
Supertransients ....................................... 321
9.4 Crises in Spatiotemporal Dynamical Systems ............ 323
9.5 Fractal Properties of Supertransients ................. 329
9.6 Turbulence in Pipe Flows .............................. 333
9.7 Closing Remarks ....................................... 338
Part IV Applications of Transient Chaos
10 Chaotic Advection in Fluid Flows ........................... 343
10.1 General Setting of Passive Advective Dynamics ......... 344
10.2 Passive Advection in von KaYman Vortex Streets ........ 346
10.3 Point Vortex Problems ................................. 351
10.4 Dye Boundaries ........................................ 358
10.5 Advection in Aperiodic Flows .......................... 362
10.6 Advection in Closed Flows with Leaks .................. 370
10.7 Advection of Finite-Size Particles .................... 373
10.8 Reactions in Open Flows ............................... 377
11 Controlling Transient Chaos and Applications ............... 385
11.1 Controlling Transient Chaos: General Introduction ..... 386
11.2 Maintaining Chaos: General Introduction ............... 392
11.3 Voltage Collapse and Prevention ....................... 395
11.4 Maintaining Chaos to Prevent Species Extinction ....... 399
11.5 Maintaining Chaos in the Presence of Noise, Safe
Sets .................................................. 405
11.6 Encoding Digital Information Using Transient Chaos .... 407
12 Transient Chaotic Time-Series Analysis ..................... 413
12.1 Reconstruction of Phase Space ......................... 414
12.2 Detection of Unstable Periodic Orbits ................. 421
12.3 Computation of Dimension .............................. 426
12.4 Computing Lyapunov Exponents from Transient
Chaotic Time Series ................................... 430
Final Remarks ................................................. 435
A Multifractal Spectra ....................................... 437
A.l Definition of Spectra ................................. 437
A.2 Multifractal Spectra for Repellers of One-
Dimensional Maps ...................................... 437
A.3 Multifractal Spectra of Saddles of Two-Dimensional
Maps .................................................. 441
A.4 Zeta Functions ........................................ 442
В Open Random Baker Maps ..................................... 445
В.1 Single Scale Baker Map ................................ 445
B.2 General Baker Map ..................................... 447
С Semiclassical Approximation ................................ 449
C.1 Semiclassical S-Matrix in Action-Angle
Representation ........................................ 449
C.2 Stationary Phase Approximation and the Maslov Index ... 450
D Scattering Cross Sections .................................. 455
D.1 Scattering Cross Sections in Classical Chaotic
Scattering ............................................ 455
D.2 Semiclassical Scattering Cross Sections ............... 457
References .................................................... 459
Index ......................................................... 491
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