Seminars & Colloquia Calendar

Abstract: We survey computational methods for approximating the global long-term behavior of dynamical systems.  We will see how to approximate sets of states which remain invariant under the dynamics and how to use these approximations in order to prove the existence of symbolic dynamics. We will then construct a linear operator which, similar to the Koopman operator, describes the evolution of probability densities on the state space.  The eigenfunctions of this operator will reveal certain macroscopic features of the dynamics.  For finite-time and time-varying dynamical systems this concept can be adapted and leads to a method for the computation of advective coherent sets (e.g. eddies in an unsteady fluid flow). We will see how these sets can actually be computed from sparse and incomplete trajectory data alone. Throughout the talk, the mathematical concepts will be illustrated by computational examples.