Hunting French Ducks in a Noisy Environment

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

1 MAPMO - Mathématiques appliquées et physique mathématique d'Orléans
2 Fakultät für Mathematik = Faculty of Mathematics [Bielefeld]
3 MPI-PKS - Max Planck Institute for the Physics of Complex Systems

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.

Mots clés: Mixed-mode oscillation; Canard; Folded node; Concentration of sample paths; First-exit time; Random dynamical system; Singular perturbation; Fast-slow system; Invariant manifold; Dynamic bifurcation

Domaines: Systèmes dynamiques [math.DS]; Probabilités [math.PR]; Neurobiologie

Nils Berglund, Barbara Gentz, Christian Kuehn. Hunting French Ducks in a Noisy Environment. Journal of Differential Equations, 2012, 252 (9), pp.4786-4841. ⟨10.1016/j.jde.2012.01.015⟩. ⟨hal-00535928⟩