Computer assisted error bounds for linear approximation of (un)stable manifolds and rigorous validation of higher dimensional transverse connecting orbits

This paper presents a method for computing validated error bounds on the truncation error associated with the linear approximation of an (un)stable invariant manifold by its eigenspace. The method is based on studying a certain functional equation which describes a chart map for the invariant manifold. The truncation error is represented as a bounded analytic function on an explicitly given neighborhood. Moreover studying this functional equation leads to a method for bounding derivatives of the truncation error as well. The methods developed in the present work are well suited for studying higher dimensional invariant manifolds, and validated numerical results are provided for manifolds of dimension up to one hundred. As an application of these ideas we present some computer assisted proofs of transverse homoclinic connecting orbits.