A Surface of Heteroclinic Connections Between Two Saddle Slow Manifolds in the Olsen Model

Abstract

The Olsen model for the biochemical peroxidase-oxidase reaction has a parameter regime where one of its four variables evolves much slower than the other three. It is characterized by the existence of periodic orbits along which a large oscillation is followed by many much smaller oscillations before the process repeats. We are concerned here with a crucial ingredient for such mixed-mode oscillations (MMOs) in the Olsen model: a surface of connecting orbits that is followed closely by the MMO periodic orbit during its global, large-amplitude transition back to another onset of small oscillations. Importantly, orbits on this surface connect two one-dimensional saddle slow manifolds, which exist near curves of equilibria of the limit where the slow variable is frozen and acts as a parameter of the so-called fast subsystem. We present a numerical method, based on formulating suitable boundary value problems, to compute such a surface of connecting orbits. It involves a number of steps to compute the slow manifolds, certain submanifolds of their stable and unstable manifolds and, finally, a first connecting orbit that is then used to sweep out the surface by continuation. If it exists, such a surface of connecting orbits between two one-dimensional saddle slow manifolds is robust under parameter variations. We compute and visualize it in the Olsen model and show how this surface organizes the global return mechanism of MMO periodic orbits from the end of small oscillations back to a region of phase space where they start again.