Generalized Cell Mapping for Randomly Perturbed Dynamical Systems

Abstract

We present a method to analyze dynamical systems undergoing random perturbations based on the cell mapping approach. Analytical expressions are derived for the transition probabilities from the evolution operator of the system. Thus there is no need for simulation of randomness and the numerical approximations are safe, i.e., we approximate attractors and their basins from within and give lower bounds for exit times. For additively perturbed systems the transition probabilities can be expressed in terms of the transition probabilities of the unperturbed system and the properties of the perturbation. The numerical details concerning the perturbation terms are discussed and their application is shown with an example.