Computing One-Dimensional Global Manifolds of Poincaré Maps by Continuation

We present an algorithm for computing one-dimensional stable and unstable manifolds of saddle periodic orbits in a Poincare section. The computation is set up as a boundary value problem by restricting both end points of orbit segments to the section. Starting from the periodic orbit itself, we use collocation routines from {\sc Auto} to continue the solutions of the boundary value problem such that one end point of the orbit segment varies along a part of the manifold that was already computed. In this way, the other end point of the orbit segment traces out a new piece of the manifold.As opposed to standard methods that use shooting to compute the Poincare map as the kth return map, our approach defines the Poincare map as the solution of a boundary value problem. This enables us to compute global manifolds through points where the flow is tangent to the section---a situation that is typically encountered unless one is dealing with a periodically forced system. Another major advantage of our approach is...