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. This paper has been revised in September 1999.
|Publication status||Unpublished - 1999|