Michael Dellnitz

Set Oriented Numerical Methods for Dynamical Systems

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

• invariant sets themselves, with particular attention to invariant manifolds;
• (natural) invariant measures;
• almost invariant sets;
• Lyapunov exponents;
• different types of dimension.

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

• identification of conformations in molecular dynamics;
• computation of zeros of functions with an application to the numerical investigation of electrical circuits;
• approximation of invariant manifolds of periodic orbits in the restricted three body problem together with an explanation in which way these results can be used in the mission design for spacecrafts;
• detection of so-called blowout bifurcations.

Michael Dellnitz
Department of Mathematics and Computer Science
University of Paderborn
D-33095 Paderborn
Germany

Oct 27 1998