Set oriented numerical methods for dynamical systems

Prof. Dr. Michael Dellnitz, University of Paderborn, Germany, Department of Mathematics and Computer Science

Frequently dynamical systems exhibit complicated temporal behavior. In this case it can be useful to approximate corresponding characterizing statistical quantities such as Lyapunov exponents or invariant measures. For this purpose there have been proposed reliable numerical techniques, which are based on a set-oriented approach rather than using single long term simulations of the underlying dynamical system. Here we give an overview about recent developments in this area.

In particular, we explain how to approximate a natural invariant measure, that is, an SRB-measure if such a measure exists. These measures provide the information about the frequency by which a typical solution is observed in different parts of state space. The numerical approximation is obtained by adaptive multilevel subdivision strategies based on box coverings of the corresponding invariant set.

In addition to the stationary statistical behavior obtained by the invariant measure we show how to approximate almost invariant sets. These are regions in state space where typical solutions stay for a relatively long period of time before leaving again.

There are several convergence results for the (adaptive) numerical methods which are presented here, e.g. the convergence of approximating measures to an SRB-measure. Additionally we justify the techniques for the computation of almost invariant sets by recent analytical results concerning the spectrum of the Perron-Frobenius operator. We illustrate both analytical results and numerical methods by several examples.