Hunting French Ducks in a Noisy Environment

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.

60 pages, 9 figures

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• On the noise-induced passage through an unstable periodic orbit II: General case
• Coherence resonance in neuronal populations: mean-field versus network model
• Early-Warning Signs for Pattern-Formation in Stochastic Partial Differential Equations
• From random Poincaré maps to stochastic mixed-mode-oscillation patterns
• Deterministic continutation of stochastic metastable equilibria via Lyapunov equations and ellipsoids
• Homogenization of Fully-Coupled Chaotic Fast-Slow Systems via Intermediate Stochastic Regularization
• Stochasticity induced mixed-mode oscillations and distribution of recurrent outbreaks in an ecosystem
• Spectral theory for random Poincaré maps
• Noise-induced canard and mixed-mode oscillations in large stochastic networks with multiple timescales
• A General View on Double Limits in Differential Equations
• Towards a new classification of bursting patterns: review & extensions