Center Manifolds for Homoclinic Solutions

In this article, center-manifold theory is developed for homoclinic solutions of ordinary differential equations or semilinear parabolic equations. A center manifold along a homoclinic solution is a locally invariant manifold containing all solutions which stay close to the homoclinic orbit in phase space for all times. Therefore, as usual, the low-dimensional center manifold contains the interesting recurrent dynamics near the homoclinic orbit, and a considerable reduction of dimension is achieved. The manifold is of class C ^1, for some >0. As an application, results of Shilnikov about the occurrence of complicated dynamics near homoclinic solutions approaching saddle-foci equilibria are generalized to semilinear parabolic equations.