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

Nils Berglund (1, 2) , Barbara Gentz (3) , Christian Kuehn (4)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 FDP - Fédération de recherche Denis Poisson
3 Fakultät für Mathematik = Faculty of Mathematics [Bielefeld]
4 Institute for Analysis and Scientific Computing [Wien]

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.

Mots clés: first-exit time; random dynamical system; return map; mixed-mode oscillation; folded node; canard; Singular perturbation; fast-slow system; dynamic bifurcation; concentration of sample paths; Markov chain

Domaines: Probabilités [math.PR]

Nils Berglund, Barbara Gentz, Christian Kuehn. From random Poincaré maps to stochastic mixed-mode-oscillation patterns. Journal of Dynamics and Differential Equations, 2015, 27 (1), pp.83-136. ⟨10.1007/s10884-014-9419-5⟩. ⟨hal-00921881v2⟩