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

TL;DR

This paper analyzes how small random perturbations cause systems with unstable and stable periodic orbits to exhibit a cycling phenomenon in their exit locations, revealing a universal distribution pattern related to the Gumbel distribution.

Contribution

It demonstrates that for a broad class of systems, the distribution of exit locations follows a universal periodicized Gumbel distribution, extending previous understanding of noise-induced transitions.

Findings

The cycling profile is universally described by a periodicized Gumbel distribution.

The distribution of first-exit locations shifts logarithmically with noise intensity.

The methods combine large deviation theory with properties of random Poincaré maps.

Abstract

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 Poincar\'e maps described by continuous-space discrete-time Markov chains.