Point-to-periodic and Periodic-to-periodic Connections

In this work we consider computing and continuing connecting orbits in parameter dependent dynamical systems. We give details of algorithms for computing connections between equilibria and periodic orbits, and between periodic orbits. The theoretical foundation for these techniques is given by the seminal work of Beyn [5] where a numerical technique is also proposed. Our algorithms consist of splitting the computation of the connection from that of the periodic orbit(s). To set up appropriate boundary conditions, we follow the algorithmic approach used in [9] for the case of connecting orbits between equilibria, and construct and exploit the smooth block Schur decomposition of the monodromy matrices associated to the periodic orbits. Numerical examples illustrate the performance of the algorithms.