berglund2012hunting
Hunting French ducks in a noisy environment
Nils Berglund, Barbara Gentz and Christian Kuehn
Journal of 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.
Figure 1: Singular limit $$\epsilon = 0$$ for the normal form (3.8) with $$\mu = 0.15$$. The attracting manifold $$C_0^a$$ (red), the fold line $$L$$ (green) and the repelling manifold $$C_0^r$$ (blue) partition the critical manifold. Trajectories of the slow (fast) subsystem are indicated by single (double) arrows.