On the Noise-Induced Passage through an Unstable Periodic Orbit II: General Case

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 largedeviation results with properties of random Poincaré maps described by continuousspace discrete-time Markov chains. Date. August 13, 2012. Revised, July 24, 2013. 2010 Mathematical Subject Classification. 60H10, 34F05 (primary), 60J05, 60F10 (secondary)