On the noise-induced passage through an unstable periodic orbit II: General case

Nils Berglund (1) , Barbara Gentz (2)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 Fakultät für Mathematik = Faculty of Mathematics [Bielefeld]

Consider a dynamical system given by a planar differential equation, which exhibits an unstable periodic orbit surrounding a stable periodic orbit. It is known that under random perturbations, the distribution of locations where the system's first exit from the interior of the unstable orbit occurs, typically displays the phenomenon of cycling: The distribution of first-exit locations is translated along the unstable periodic orbit proportionally to the logarithm of the noise intensity as the noise intensity goes to zero. We show that for a large class of such systems, the cycling profile is given, up to a model-dependent change of coordinates, by a universal function given by a periodicised Gumbel distribution. Our techniques combine action-functional or large-deviation results with properties of random Poincaré maps described by continuous-space discrete-time Markov chains.

Mots clés: Synchronisation; Large deviations; Limit cycle; Stochastic resonance; Phase slip; Cycling; Stochastic exit problem; Diffusion exit; First-exit time; Characteristic boundary; Gumbel distribution

Domaines: Probabilités [math.PR]

Nils Berglund, Barbara Gentz. On the noise-induced passage through an unstable periodic orbit II: General case. SIAM Journal on Mathematical Analysis, 2014, 46 (1), pp.310-352. ⟨10.1137/120887965⟩. ⟨hal-00723729v2⟩