Hunting French Ducks in a Noisy Environment

Details Hunting French Ducks in a Noisy Environment Hunting French Ducks in a Noisy Environment

2010-11-14

arxiv.org/abs/1011.3193v1

J. Differential Equations 252 (2012) 4786-4841

Mathematics

Dynamical Systems

60 pages, 9 figures

Scientific paper

10.1016/j.jde.2012.01.015

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 above 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.

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