Many interesting characterizing quantities of dynamical systems are of statistical nature. These include Lyapunov exponents, the measure theoretic entropy or certain types of dimension but also (natural) invariant measures themselves provide important statistical information on the dynamical behavior. The first step in a numerical computation of these quantities is to approximate the corresponding invariant set. This can be done by direct simulation or by a set oriented approach. We choose the second possibility and show how to construct corresponding algorithms based on (adaptive) multilevel subdivision techniques. The underlying idea is to approximate the (deterministic) dynamical behavior by an appropriate Markov chain. Then the desired statistical information can be extracted from this stochastic process.
Concretely we present algorithms for the computation of
We discuss both theoretical properties of these algorithms (e.g. convergence results) and aspects concerning the numerical realization (e.g. adaptive schemes, parallelization strategies). Moreover applications of these methods to different fields are indicated, in particular the
Department of Mathematics and Computer Science
University of Paderborn
Oct 27 1998