| 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 chaotic 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 paradigmatic flow system: a cylinder in a two-dimensional incompressible flow, behind which von Karman 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. |
