Center manifolds for homoclinic solutions
- Sandstede, Björn
2010 Mathematics Subject Classification
- 34C45 34C37 34C23
- center manifolds, Shilnikov bifurcation, homoclinic solutions, saddle-foci equilibria, semilinear parabolic equations
In this article, center-manifold theory for homoclinic solutions of ordinary differential equations or semilinear parabolic equations is developed. Here, a center manifold along a homoclinic orbit q(t) is a locally invariant manifold containing all solutions which stay close to q(t) in phase space for all times. Therefore, as usual, the low-dimensional center manifold contains the interesting recurrent dynamics nearby the homoclinic orbit and a considerable reduction of dimension is achieved. The manifold is of class C1,β for some β > 0. As one application, results of Shilnikov about the occurrence of complicated dynamics nearby homoclinic solutions approaching saddle-foci equilibria are generalized to semilinear parabolic equations.