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December  2009, 2(4): 807-827. doi: 10.3934/dcdss.2009.2.807

The geometry of mixed-mode oscillations in the Olsen model for the Peroxidase-Oxidase reaction

Mathieu Desroches 1, , Bernd Krauskopf 1, and  Hinke M. Osinga 1,

1. 

Bristol Center for Applied Nonlinear Mathematics, Department of Engineering Mathematics, University of Bristol, Queen's Building, Bristol BS8 1TR, United Kingdom, United Kingdom, United Kingdom

Received  September 2008 Revised  February 2009 Published  September 2009

We study the organization of mixed-mode oscillations (MMOs) in the Olsen model for the peroxidase-oxidase reaction, which is a four-dimensional system with multiple time scales. 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 associated with 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.
Keywords: delayed Hopf bifurcation, mixed-mode oscillations, invariant manifolds..
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Mathieu Desroches, Bernd Krauskopf, Hinke M. Osinga. The geometry of mixed-mode oscillations in the Olsen model for the Peroxidase-Oxidase reaction. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 807-827. doi: 10.3934/dcdss.2009.2.807
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