On the Approximation of Complicated Dynamical Behavior

We present efficient techniques for the numerical approximation of complicated dynamical behavior. In particular, we develop numerical methods which allow to approximate SRB-measures as well as (almost) cyclic behavior of a dynamical system. The methods are based on an appropriate discretization of the Frobenius-Perron operator, and two essentially different mathemat-ical concepts are used: the idea is to combine classical convergence results for finite dimensional approximations of compact operators with results from Ergodic Theory concerning the approxima-tion of SRB-measures by invariant measures of stochastically perturbed systems. The efficiency of the methods is illustrated by several numerical examples. Key words. computation of invariant measures, approximation of the Frobenius-Perron oper-ator, computation of SRB-measures, almost invariant set, cyclic behavior AMS subject classifications. 58F11, 65L60, 58F12