About

We are interested in the effect of weak noise on ODEs that are either periodically forced in time, or admit several periodic orbits. Such equations arise in a number of applications, including synchronisation, stochastic resonance, and models for membrane potential dynamics in neuroscience. This research is partly funded by the ANR project PERISTOCH.

Noise-induced passage through an unstable periodic orbit

In this and this paper, we studied two-dimensional SDEs with two stable, and one unstable periodic orbit separating the two stable ones. The main result characterises the distribution of the first-passage location through the unstable periodic orbit. This distribution can be decomposed as a sum of a shift depending logarithmically on the noise intensity, an asymptotically geometric random variable, and a variable following a Gumbel distribution. The logarithmic dependence on the noise intensity is a manifestation of the phenomenon of cycling, described here in more detail. See here for an application to residence-time distributions in stochastic resonance, and here for an application to phase slips in synchronisation, and links with extreme-value theory.

This paper sharpens the results in this one by determining Eyring-Kramers type asymptotics of the tranisition time.

Random Poincaré maps

Several applications have made apparent the need for a theory of random Poincaré maps, describing successive intersections of sample paths of an SDE with a surface of section, separated by a suitably defined excursion: the study of the noise-induced passage through an unstable periodic orbit made here, the analysis of interspike interval statistics in the stochastic FitzHugh-Nagumo model carried out here, and the study of mixed-mode oscillations near a folded-node bifurcation performed here. This paper carries out a first step towards a general theory of random Poincaré maps, by characterising eigenvalues exponentially close to 1 of these maps, and the associated eigenfunctions.

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