Cell Mapping applied to Random Dynamical Systems

Abstract

The method of Cell Mapping is a numerical tool to analyze the long-term behavior of dynamical systems. For deterministic systems, the notion was first introduced by HSU and it is shown by GUDER AND KREUZER that Cell Mapping in this context represents an approximation of the Frobenius-Perron operator by a Galerkin method. Our purpose is to extend the concept to randomly perturbed dynamical systems or, more general, to Random Dynamical Systems. We show that time evolution of absolutely continuous measures and the corresponding densities can be described by Markov operators whose fixed points refer to stationary measures and densities, respectively. A projection of the Markov operator on densities onto a discrete basis set of characteristic functions leads directly to the reformulation of Cell Mapping against the background of stochastic dynamics. © 2007 Springer.