We study the organization of mixed-mode oscillations (MMOs) in the Olsen model for peroxidase-oxidase reaction. This model is a four-dimensional slow-fast system, but it does not have a clear split into slow and fast variables. A numerical continuation study shows that the MMOs appear as families in a complicated bifurcation structure that involves many regions of multistability. We show that the small-amplitude oscillations of the MMOs arise from the slow passage through a (delayed) Hopf bifurcation of a three-dimensional fast subsystem, while large-amplitude excursions are due to a global reinjection mechanism. To characterize these two key components of MMO dynamics geometrically we consider attracting and repelling slow manifolds in phase space. More specifically, these objects are surfaces that are defined and computed as one-parameter families of stable and unstable manifolds of saddle equilibria of the fast subsystem. The attracting and repelling slow manifolds interact near the Hopf bifurcation, but also explain the geometry of the global reinjection mechanism. Their intersection gives rise to canard-like orbits that organize the spiralling nature of the MMOs.
|Publication status||Unpublished - 12 Sep 2008|
- invariant manifolds
- mixed-mode oscillations
- delayed Hopf bifurcation