From Random Poincaré Maps to Stochastic Mixed-Mode-Oscillation Patterns

  • Nils Berglund
  • Barbara Gentz
  • Christian KuehnEmail author


We quantify the effect of Gaussian white noise on fast–slow dynamical systems with one fast and two slow variables, which display mixed-mode oscillations owing to the presence of a folded-node singularity. The stochastic system can be described by a continuous-space, discrete-time Markov chain, recording the returns of sample paths to a Poincaré section. We provide estimates on the kernel of this Markov chain, depending on the system parameters and the noise intensity. These results yield predictions on the observed random mixed-mode oscillation patterns. Our analysis shows that there is an intricate interplay between the number of small-amplitude oscillations and the global return mechanism. In combination with a local saturation phenomenon near the folded node, this interplay can modify the number of small-amplitude oscillations after a large-amplitude oscillation. Finally, sufficient conditions are derived which determine when the noise increases the number of small-amplitude oscillations and when it decreases this number.


Fast–slow system Folded node Mixed–mode oscillation Random dynamical system Concentration of sample paths Markov chain 

Mathematics Subject Classification

Primary 37H20 34E17 Secondary 60H10 



C.K. would like to thank the Austrian Academy of Sciences (ÖAW) for support via an APART fellowship as well as the European Commission (EC/REA) for support by a Marie-Curie International Re-integration Grant. B.G. and C.K. thank the MAPMO at the Université d’Orléans, N.B. and C.K. thank the CRC 701 at the University of Bielefeld for kind hospitality and financial support. Last but not least, we would like to thank an anonymous referee for constructive comments that led to improvements in the presentation. Research supported by CRC 701 “Spectral Structures and Topological Methods in Mathematics”.


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Authors and Affiliations

  1. 1.MAPMO, CNRS – UMR 7349, Université d’Orléans and FédérationOrléans Cedex 2France
  2. 2.Faculty of MathematicsUniversity of BielefeldBielefeldGermany
  3. 3.Institute for Analysis and Scientific ComputingVienna University of TechnologyViennaAustria

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