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We present efficient techniques for the numerical approximation of complicated dynamical behavior. In particular, we develop numerical methods which allow us to approximate Sinai-Ruelle-Bowen (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 mathematical concepts are used: our idea is to combine classical convergence results for finite dimensional approximations of compact operators with results from ergodic theory concerning the approximation 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 operator, computation of SRB-measures, almost invariant set, cyclic behavior that is, the dynamics modulo complex (unpredictable) behavior which is due to the presence of chaos.From a practical point of view the most important invariant measures are the socalled SRB-measures. The reason is that for these measures the spatial and temporal averages of observables are identical for a set of initial conditions which has positive Lebesgue measure. The introduction of the underlying concept goes back to Y. Sinai (see [27]), and the existence of these measures has been shown for Axiom A systems by D. Ruelle and R. Bowen [26,3]. In this article we suggest a numerical method for the approximation of SRB-measures, and in this context their stochastic stability is particularly important: first we use this fact as an analytical tool in our main convergence result in section 4, and second it is of practical importance if we view the numerical approximation as a small random perturbation. Indeed, stochastic stability of SRB-measures is guaranteed for Axiom A systems [17,18].More precisely there are two essential mathematical ingredients which allow us to develop a numerical method for the approximation of SRB-measures. We use a result of Yu. Kifer on the convergence of invariant measures in stochastic perturbations of the underlying dynamical system to the SRB-measure (see [17]) and combine this with results on the convergence of eigenspaces of discretized compact operators (see, e.g., [25]). The same technique is used for the approximation of the subsets in state space which are (almost) cyclically permuted by the dynamical process. With respect to the approximation of SRB-measures a similar result has previously been obtained by F. Hunt (see [15]). However, our methods are quite different from the ones used in that work. In particular, the results stated here cover the important situation where the random perturbations have a probability distribution with local support. In fact, this is the relevant case having in mind that the round off error in the numerical approximation can be interpreted as such a local perturbation. Another approach for the computation of SRB-measures-avoiding the approximation of the Frobeniu...

Each invariant set of a given dynamical system is part of the global attractor. Therefore the global attractor contains all the potentially interesting dynamics, and, in particular, it contains every (global) unstable manifold. For this reason it is of interest to have an algorithm which allows to approximate the global attractor numerically. In this article we develop such an algorithm using a subdivision technique. We prove convergence of this method in a very general setting, and, moreover, we describe the qualitative convergence behavior in the presence of a hyperbolic structure. The algorithm can successfully be applied to dynamical systems of moderate dimension, and we illustrate this fact by several numerical examples. Classification (1991): 65L05, 65L70, 58F15, 58F12 Mathematics Subject

The behaviors of trajectories of nonlinear dynamical systems are notoriously hard to characterize and predict. Rather than characterizing dynamical behavior at the level of trajectories, we consider following the evolution of sets. There are often collections of sets that behave in a very predictable way, in spite of the fact that individual trajectories are entirely unpredictable. Such special collections of sets are invisible to studies of long trajectories. We describe a global set-oriented method to detect and locate these large dynamical structures. Our approach is a marriage of new ideas in modern dynamical systems theory and the novel application of graph dissection algorithms.

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The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable manifold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example — the two-dimensional stable manifold of the origin in the Lorenz system.

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