Hunting French Ducks in a Noisy Environment

TL;DR

This paper investigates how Gaussian white noise impacts mixed-mode oscillations in fast-slow dynamical systems with a folded-node, revealing critical noise levels that obscure small oscillations and induce early jumps away from canard solutions.

Contribution

It provides a quantitative analysis of noise effects on mixed-mode oscillations and proves that noise can cause early jumps, altering system dynamics.

Findings

Critical noise intensities hide small oscillations.

Noise causes high-probability jumps from canard solutions.

Early jumps significantly change oscillation patterns.

Abstract

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.