Dynamics, Topology, and Computation

Konstantin Mischaikow, Georgia Institute of Technology

Our ability to numerically simulate the dynamics of nonlinear systems easily exceeds our ability to rigorously verify (or deny) the observed results. This issue is of particular importance in chaotic systems where sensitivity to initial conditions implies that, due to roundoff errors, the validity of any single numerically computed trajectory is at best suspect, and in infinite dimensional systems where the simulation can only be performed on an approximated system of equations. Using a variety of examples, including the Kuramoto-Sivashinsky, Swift-Hohenberg, Cahn-Hilliard, and Kot-Schaffer equations, I will discuss recently developed computationally efficient techniques, based on the Conley index, that allow us to rigorously prove the existence of fixed points, connecting orbits, periodic orbits, and chaotic dynamics.