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We study the Olsen model for the peroxidase-oxidase reaction. The dynamics isanalyzed using a geometric decomposition based upon multiple time scales. TheOlsen model is four-dimensional, not in a standard form required by geometricsingular perturbation theory and contains multiple small parameters. Thesethree obstacles are the main challenges we resolve by our analysis. Scaling andthe blow-up method are used to identify several subsystems. The resultspresented here provide a rigorous analysis for two oscillatory modes. Inparticular, we prove the existence of non-classical relaxation oscillations intwo cases. The analysis is based upon desingularization of lines oftranscritical and submanifolds of fold singularities in combination with anintegrable relaxation phase. In this context our analysis also explains anassumption that has been utilized, based purely on numerical reasoning, in aprevious bifurcation analysis by Desroches, Krauskopf and Osinga[{Discret.}{Contin.}{Dyn.}{Syst.}S, 2(4), p.807--827, 2009]. Furthermore, thegeometric decomposition we develop forms the basis to prove the existence ofmixed-mode and chaotic oscillations in the Olsen model, which will be discussedin more detail in future work.

Multiscale Geometry of the Olsen Model and Non-Classical Relaxation Oscillations