Computing One-Dimensional Global Manifolds of Poincaré Maps by Continuation

We present an algorithm to compute one-dimensional stable and unstable manifolds of saddle periodic orbits in a Poincaré section. The computation is set up as a boundary value problem by restricting the beginning and end points of orbit segments to the section. Starting from the periodic orbit itself, we use collocation routines from 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 Poincaré map as the k-th return map, our approach defines the Poincaré map as the solution to 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 that it deals effectively with the problem of extreme sensitivity of the Poincaré map on its argument, which is a typical feature in the important class of slow-fast systems. We illustrate and test our algorithm by computing stable and unstable manifold in three examples: the forced Van der Pol oscillator, a model of a semiconductor laser with optical injection, and a slow-fast chemical oscillator. All examples are accompanied by animations of how the manifolds grow during the computation.