Chapter 5 - Set Oriented Numerical Methods for Dynamical Systems

This chapter focuses on set oriented numerical methods for dynamical systems. The set oriented numerical methods can be used to approximate different types of invariant sets or invariant manifolds but they also allow extracting statistical information on the dynamical behavior via the computation of natural invariant measures or almost invariant sets. In contrast to other numerical techniques, these methods do not rely on the computation of single long-term trajectories but rather use the information obtained from several short-term trajectories. Set oriented method can also be used for the computation of invariant manifolds. Although the method can, in principle, be applied to manifolds of arbitrary hyperbolic invariant sets. An important statistical characterization of the behavior of a dynamical system is given by so-called SRB (Sinai–Ruelle–Bowen) measures. The important property of these invariant measures is that they lend weight to a region in phase space according to the probability by which typical trajectories visit this region.

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