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.
Publié le : 2012-05-01
Classification:
Concentration of sample paths,
First-exit time,
Random dynamical system,
Singular perturbation,
Fast-slow system,
Invariant manifold,
Dynamic bifurcation,
Folded node,
Canard,
Mixed-mode oscillation,
MSC 37H20, 34E17 (primary), 60H10 (secondary),
[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS],
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR],
[SDV.NEU.NB]Life Sciences [q-bio]/Neurons and Cognition [q-bio.NC]/Neurobiology
@article{hal-00535928,
author = {Berglund, Nils and Gentz, Barbara and Kuehn, Christian},
title = {Hunting French Ducks in a Noisy Environment},
journal = {HAL},
volume = {2012},
number = {0},
year = {2012},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00535928}
}
Berglund, Nils; Gentz, Barbara; Kuehn, Christian. Hunting French Ducks in a Noisy Environment. HAL, Tome 2012 (2012) no. 0, . http://gdmltest.u-ga.fr/item/hal-00535928/