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

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 Poincaré maps described by continuous-space discrete-time Markov chains.

44 pages, 4 figures

References in corpus (4)

• The Morris-Lecar neuron model embeds a leaky integrate-and-fire model
• On the noise-induced passage through an unstable periodic orbit I: Two-level model
• Universality of residence-time distributions in non-adiabatic stochastic resonance
• Suppression of thermally activated escape by heating

Cited by in corpus (19)

• Generalisation of the Eyring-Kramers transition rate formula to irreversible diffusion processes
• On the noise-induced passage through an unstable periodic orbit I: Two-level model
• Stochastic Bifurcations induced by correlated Noise in a Birhythmic van der Pol System
• Universality of residence-time distributions in non-adiabatic stochastic resonance
• From random Poincaré maps to stochastic mixed-mode-oscillation patterns
• Minimum Action Path theory reveals the details of stochastic biochemical transitions out of oscillatory cellular states
• Synchronization of stochastic mean field networks of Hodgkin-Huxley neurons with noisy channels
• Stochastic resonance in stochastic PDEs
• A Dynamical Systems Approach for Most Probable Escape Paths over Periodic Boundaries
• Spectral theory for random Poincaré maps
• An Eyring-Kramers law for slowly oscillating bistable diffusions
• Reducing metastable continuous-space Markov chains to Markov chains on a finite set
• Noise-induced phase slips, log-periodic oscillations, and the Gumbel distribution
• Stochastic Resonance for a Model with Two Pathways
• A comparative study of stochastic resonance for a model with two pathways by escape times, linear response, invariant measures and the conditional Kolmogorov-Smirnov Test
• Long-time dynamics of stochastic differential equations
• Exit event from a metastable state and Eyring-Kramers law for the overdamped Langevin dynamics
• Hyperbolic periodic orbits in nongradient systems and small-noise-induced metastable transitions
• On Gumbel limit for the length of reactive paths