Escape from attracting sets in randomly perturbed systems

METASTABILITY, LYAPUNOV EXPONENTS, ESCAPE RATES, AND TOPOLOGICAL ENTROPY IN RANDOM DYNAMICAL SYSTEMS

We explore the concept of metastability in random dynamical systems, focusing on connections between random Perron–Frobenius operator cocycles and escape rates of random maps, and on topological entropy of random shifts of finite type. The Lyapunov spectrum of the random Perron–Frobenius cocycle and the random adjacency matrix cocycle is used to decompose the random system into two disjoint random systems with rigorous upper and lower bounds on (i) the escape rate in the setting of random maps, and (ii) topological entropy in the setting of random shifts of finite type, respectively.

Fractal snapshot components in chaos induced by strong noise

In systems exhibiting transient chaos in coexistence with periodic attractors, the inclusion of weak noise might give rise to noise-induced chaotic attractors. When the noise amplitude exceeds a critical value, an extended attractor appears along the fractal unstable manifold of the underlying nonattracting chaotic set. A further increase of noise leads to a fuzzy nonfractal pattern. By means of the concept of snapshot attractors and random maps, we point out that the fuzzy pattern can be decomposed into well-defined fractal components, the snapshot attractors belonging to a given realization of the noise and generated by following an ensemble of noisy trajectories. The pattern of the snapshot attractor and its characteristic numbers, such as the finite time Lyapunov exponents and numerically evaluated fractal dimensions, change continuously in time. We find that this temporal fluctuation is a robust property of the system which hardly changes with increasing ensemble size. The validity of the Kaplan-Yorke formula is also investigated. A superposition of about 100 snapshot attractors provides a good approximant to the fuzzy noise-induced attractor at the same noise strength.