Computational methods for global dynamics

We survey computational methods for approximating the global long term behavior of dynamical systems. At the core, these methods are based on an operator, the transfer operator, which describes how probability densities on state space evolve under the dynamics. Starting with a set-oriented approach for computing arbitrary invariant sets, we describe how to use the resulting covering in order to obtain a finite matrix description of the transfer operator. From its spectrum, cyclic and almost invariant (metastable) macroscopic dynamics can be detected.

In the case of a differential equation, there is an entire semigroup of these operators and it suffices to approximate its generator in order to obtain a macroscopic description of the long term dynamics - without any trajectory integration. For time-varying systems, so called coherent sets can also be computed by a seemingly different approach based on geometric ideas, leading to a finite-element based computational method which also works for sparse and incomplete trajectory data. Throughout the talk, the mathematical concepts will be illustrated by computational examples.