Flow Box Tiling Methods for Compact Invariant Manifolds

Invariant manifolds are important to the study of the qualitative behavior of dynamical systems and nonlinear control. Good algorithms exist for finding many one dimensional invariant curves, such as periodic orbits, orbits connecting fixed points, and more recently two dimensional stable and unstable manifolds of fixed points. This paper addresses the problem of computing higher dimensional closed invariant manifolds and manifolds with more complicated flows, using an approach that produces a large system of coupled two point boundary value problems. The algorithm described here is not limited to a particular dimension or topology. It does not assume that a closed global section exists, nor that a splitting or parameterization of the manifold is known a priori. A flow box tiling is used to construct a set of trajectory fragments on the manifold which are used to pose a system of coupled two point boundary value problems for the manifold.