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

Christian Kuehn, Peter Szmolyan

Research output: Contribution to journalArticlepeer-review

28 Scopus citations


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.

Original languageEnglish
Pages (from-to)583-629
Number of pages47
JournalJournal of Nonlinear Science
Issue number3
StatePublished - 1 Jun 2015
Externally publishedYes


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


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