It is well-known how to globalize one-dimensional unstable manifolds for planar vector fields and maps. We consider the case of a two-dimensional manifold in a three-dimensional space. The presented algorithm is designed for the computation of the two-dimensional unstable manifold of a normally hyperbolic invariant circle of saddle-type of a three-dimensional map. We briefly discuss how to compute this invariant circle and how to obtain the starting data for the globalization.
The algorithm computes growing pieces of the unstable manifold by using a method that does not depend on the dynamics on the manifold. Also, the algorithm is such that it guarantees the quality of the mesh on this manifold. The same algorithm can be used for the globalization of a two-dimensional unstable manifold of a hyperbolic fixed point. Furthermore, we discuss how to use a similar technique for vector fields.
This is joint work with Bernd Krauskopf.