Connecting Orbits for Compact Infinite Dimensional Maps: Computer Assisted Proofs of Existence

Abstract

We develop and implement a computer assisted argument for proving the existence of heteroclinic/homoclinic connecting orbits for compact infinite dimensional maps. The argument is based on a posteriori analysis of a certain “discrete time boundary value problem,” and a key ingredient is representing the local stable/unstable manifolds of the fixed points. For a compact mapping the stable manifold is infinite dimensional, and an important component of the present work is the development of computer assisted error bounds for numerical approximation of infinite dimensional stable manifolds. As an illustration of the utility of our method we prove the existence of some connecting orbits for a nonlinear dynamical system which appears in mathematical ecology as a model of a spatially distributed ecosystem with population dispersion.