Journal
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
Volume 46, Issue 1, Pages 310-352Publisher
SIAM PUBLICATIONS
DOI: 10.1137/120887965
Keywords
stochastic exit problem; diffusion exit; first-exit time; characteristic boundary; limit cycle; large deviations; synchronization; phase slip; cycling; stochastic resonance; Gumbel distribution
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Funding
- ANR project MANDy, Mathematical Analysis of Neuronal Dynamics [ANR-09-BLAN-0008-01]
- CRC 701 Spectral Structures and Topological Methods in Mathematics
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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 periodicized Gumbel distribution. Our techniques combine action-functional or large-deviation results with properties of random Poincare maps described by continuous-space discrete-time Markov chains.
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