Covering an invariant manifold with fat trajectories

Abstract

Invariant manifolds are important objects in the study of dynamical systems, as well as several applications. They are challenging to compute because even in simple systems they can be very complicated surfaces, demanding adaptive schemes to deal with large curvatures. This paper describes a method that represents the invariant manifold as a set of circular disks in the tangent space, projected onto the manifold which overlap and cover the manifold. These disks are found by integrating fat trajectories, which add tangent and curvature information to the usual point in phase space, and integrates these quantities along a trajectory. Using a covering eliminates the usual problems with advancing front approaches, and the dual of the covering is a triangulation, should one be needed. © 2006 Springer-Verlag Berlin Heidelberg.