### 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