Multimedia supplement for the paper
Global manifolds of vector fields: The general case
by Bernd Krauskopf and Hinke Osinga
submitted to IMA J. Numerical Analysis

Abstract

For any 1 < k < n, we show how to compute the k-dimensional stable or unstable manifold of an equilibrium in a vector field with an n-dimensional phase space. The manifold is grown as concentric (topological) (k-1)-spheres, which are computed as a set of intersection points of the manifold with a finite number of hyperplanes perpendicular to the last (k-1)-sphere. These intersection points are found by solving a suitable boundary value problem. In combination with a method for adding or removing hyperplanes we ensure that the mesh that represents the computed manifold is of a prescribed quality.

As examples we compute two-dimensional stable manifolds in the Lorenz system and in a four-dimensional Hamiltonian system from optimal control theory.

What follows are colored stills and animations of manifolds for the above two examples.

	Stills and animations of a two-dimensional stable manifold in the three-dimensional Lorenz system
	Stills and animations of a two-dimensional stable manifold in a four-dimensional Hamiltonian system from optimal control