Journal of Nonlinear Science

, Volume 25, Issue 3, pp 583–629 | Cite as

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

  • Christian Kuehn
  • Peter Szmolyan


We study the Olsen model for the peroxidase–oxidase reaction. The dynamics is analyzed using a geometric decomposition based on multiple timescales. The Olsen model is four-dimensional, not in a standard form required by geometric singular perturbation theory and contains multiple small parameters. These three obstacles are the main challenges we resolve by our analysis. Scaling and the blow-up method are used to identify several subsystems. The results presented here provide a rigorous analysis for two oscillatory modes. In particular, we prove the existence of non-classical relaxation oscillations in two cases. The analysis is based on desingularization of lines of transcritical and submanifolds of fold singularities in combination with an integrable relaxation phase. In this context, our analysis also explains an assumption that has been utilized, based purely on numerical reasoning, in a previous bifurcation analysis by Desroches et al. (Discret Contin Dyn Syst S 2(4):807–827, 2009). Furthermore, the geometric decomposition we develop forms the basis to prove the existence of mixed-mode and chaotic oscillations in the Olsen model, which will be discussed in more detail in future work.


Olsen model Multiple timescales Relaxation oscillation  Geometric singular perturbation theory Blow-up method Bifurcation delay 



CK would like to thank the Austrian Academy of Sciences (ÖAW) for support via an APART fellowship. CK and PS would like to thank the European Commission (EC/REA) for support by a Marie-Curie International Re-integration Grant.


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institute for Analysis and Scientific ComputingVienna University of TechnologyViennaAustria

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