Hunting French Ducks in a Noisy Environment

Nils Berglund, Barbara Gentz and Christian Kuehn
J. Differential Equations 252:4786-4841 (2012)

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.

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

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

 

Journal Homepage Published article: 10.1016/j.jde.2012.01.015 MR2891347 Zbl1293.37025

PDF file (2 Mb) hal-00535928 arXiv/1011.3193