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Hunting French ducks in a noisy environment. (English) Zbl 1293.37025

Summary: We consider the effect of Gaussian white noise on fast-slow dynamical systems with one fast and two slow variables, containing a folded node singularity. In the absence of noise, these systems are known to display mixed-mode oscillations, consisting of alternating large- and small-amplitude oscillations. We quantify the effect of noise and obtain critical noise intensities beyond which the small-amplitude oscillations become hidden by fluctuations. Furthermore we prove that the noise can cause sample paths to jump away from so-called canard solutions with high probability before deterministic orbits do. This early-jump mechanism can drastically influence the local and global dynamics of the system by changing the mixed-mode patterns.

MSC:

37H20 Bifurcation theory for random and stochastic dynamical systems
34E17 Canard solutions to ordinary differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34E15 Singular perturbations for ordinary differential equations
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