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

Nils Berglund, Barbara Gentz and Christian Kuehn
J. Dynamics Differential Equations 27 (1):83-136 (2015)

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.

Mathematical Subject Classification: 37H20, 34E17 (primary), 60H10 (secondary).

Keywords and phrases: Singular perturbation, fast-slow system, dynamic bifurcation, folded node, canard, mixed-mode oscillation, return map, random dynamical system, first-exit time, concentration of sample paths, Markov chain.

 

Journal Homepage Published article: 10.1007/s10884-014-9419-5 MR3317393 Zbl1348.37087

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