Title: Verified Computational Methods in Dynamical Systems

Abstract: There are a number of numerical techniques which provide great insight for analyzing various dynamical systems. These include the calculation of orbits, estimation of periodic points, etc. In many applications, these calculations suggest theorems which might be true, and then often one uses traditional mathematical methods to provide rigorous proofs. In other cases, the numerically observed phenomena are beyond the currently available tools of traditional analysis. In several of these latter cases, so-called computer assisted proofs have recently been provided. We describe some of the recent development in this area, focusing on verified methods for the computation of invariant manifolds and attracting periodic points of high period in the two dimensional Henon family. We also discuss several open problems for two dimensional systems motivated by known results in one dimensional dynamics.