Chaotic Transient Behavior of Dynamical Systems under Random Perturbations

다국어 초록 (Multilingual Abstract)

This dissertation treats three problems in nonlinear dynamics: (1) reliability of numerical trajectories, (2) noise-induced superpersistent chaotic transient, and (3) stability of attractor formed by physical particles in open chaotic flows under noise. The common theme of all three problems is transient chaos.
For the first problem, shadowing dynamics, which deals with validity of numerical computations, was studied. Due to computer round-offs, a numerical trajectory from a chaotic system can remain valid only for a finite time. This is the shadowing time. The probability distribution of the shadowing time in nonhyperbolic chaotic systems with unstable dimension variability was found to exhibit a universal, algebraic scaling for short times and a non universal, exponential scaling for long times.
Secondly, the phenomenon of noise-induced superpersistent chaotic transients was investigated with focus on scaling laws for the average transient lifetime versus the noise amplitude. In the case where a chaotic attractor exists in the absence of noise, a new class of chaotic transients was found and characterized by the double exponential dependence and the algebraic divergence for small noise. For the case where there is already a superpersistent chaotic transient, noise can significantly reduce the transient lifetime, in contrast to what was previously speculated. These results add to the understanding of the interplay between random and deterministic ch aotic dynamics with surprising physical consequence and implications.
The third problem concerns the structural stability of attractors formed by inertial particles in open chaotic flows. The effect of random perturbations on attractors was studied in a paradig-matic flow system: a cylinder in a two-dimensional incompressible flow, behind which von Ka´rma´n vortex street forms. It was found that attractors can be destroyed by small, additive noise. The resulting chaotic transient dynamics were found to be superpersistent. This happens regardless of the nature of the original attractor, chaotic or nonchaotic. Because the transient occurs in the physical space, these results suggest a way to observe superpersistent chaotic transients directly in laboratory experiments.

목차 (Table of Contents)

• ABSTRACT = ⅲ
• TABLE OF CONTENTS = ⅶ
• LIST OF FIGURES = x
• CHAPTER 1 INTRODUCTION = 1
• CHAPTER 2 STATISTICS OF SHADOWING TIME IN NONHYPERBOLIC CHAOTIC SYSTEMS WITH UNSTABLE DIMENSION VARIABILITY = 9
• 2.1. Background of Chapter 2 = 9
• 2.2. Basic concepts in shadowing dynamics = 13
• 2.2.1. Hyper bolicity of chaotic invariant sets = 13
• 2.2.2. Shadowing in hyper bolic systems = 14
• 2.2.3. Shadowing in nonhyperbolic systems with tangencies = 15
• 2.2.4. Unstable dimension variability and breakdown of shadowing = 16
• 2.2.5. Shadowing lemma in nonuniform hyperbolic systems = 18
• 2.3. Procedure to compute the shadowing distance and time = 19
• 2.3.1. Pointwise shadowing distance and brittleness = 19
• 2.3.2. Test brittleness = 20
• 2.4. Numerical examples = 22
• 2.4.1. A three-dimensional map = 22
• 2.4.2. The kicked double-rotor map = 26
• 2.5. Physical theory for statistics of shadowing time = 35
• 2.6. Discussions of Chapter 2 = 39
• CHAPTER 3 SCALING LAWS FOR NOISE-INDUCED SUPERPERSISTENT CHAOTIC TRANSIENTS = 41
• 3.1. Background of Chapter 3 = 41
• 3.2. Model for noise-induced superpersistent chaotic transients = 45
• 3.2.1. Dynamical mechanism for noise-induced superpersistent chaotic transients = 45
• 3.2.2. A prototype model for superpersistent chaotic transients = 46
• 3.3. Scaling theory for average tunneling time and average chaotic transient lifetime = 50
• 3.3.1. The critical case = 51
• 3.3.2. Supercritical regime = 53
• 3.3.3. Subcritical regime = 55
• 3.3.4. Summary of scaling laws = 57
• 3.4. Numerical support = 58
• 3.4.1. Average tunneling time = 58
• 3.4.2. A two-dimensional map = 61
• 3.5. Discussions of Chapter 3 = 69
• Appendix of Chapter 3 = 71
• CHAPTER 4 STABILITY OF ATTRACTORS FORMED BY INERTIAL PARTICLES IN OPEN CHAOTIC FLOWS = 75
• 4.1. Background of Chapter 4 = 75
• 4.2. Model of advective dynamics of inertial particles = 79
• 4.2.1. Open flow model = 79
• 4.2.2. Inertial dynamics = 81
• 4.2.3. Attractors = 82
• 4.3. Noise-induced superpersistent chaotic transients = 84
• 4.3.1. Chaotic attractors = 85
• 4.3.2. Periodic attractors = 93
• 4.4. Scaling theory = 93
• 4.5. Discussions of Chapter 4 = 100
• REFERENCES = 103
• APPENDIX A BASIC CONCEPTS IN STOCHASTIC PROCESS = 111
• A.1. Stochastic process = 112
• A.2. The Fokker-Planck equation = 112
• A.3. Mean first passage time = 114
• APPENDIX B NUMERICAL METHOD = 117