From random Poincaré maps to stochastic mixed-mode-oscillation patterns

Abstract : 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.
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Journal of Dynamics and Differential Equations, Springer Verlag, 2015, 27 (1), pp.83-136. 〈http://link.springer.com/article/10.1007%2Fs10884-014-9419-5〉. 〈10.1007/s10884-014-9419-5〉
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Nils Berglund, Barbara Gentz, Christian Kuehn. From random Poincaré maps to stochastic mixed-mode-oscillation patterns. Journal of Dynamics and Differential Equations, Springer Verlag, 2015, 27 (1), pp.83-136. 〈http://link.springer.com/article/10.1007%2Fs10884-014-9419-5〉. 〈10.1007/s10884-014-9419-5〉. 〈hal-00921881v2〉

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