Oscillatory Behavior of the Rate of Escape through an Unstable Limit Cycle

TL;DR

This paper investigates how the escape rate in a two-dimensional dynamical system with an unstable limit cycle behaves under weak noise, revealing a non-Arrhenius exponential decay modulated by oscillations due to nondifferentiability.

Contribution

It demonstrates that the weak-noise escape rate includes a periodic oscillation factor in the logarithm of noise strength, caused by the nondifferentiability of the system's potential at the limit cycle.

Findings

Escape rate falls off exponentially with decreasing noise.

The escape rate includes a periodic oscillation in the logarithm of noise strength.

Implications for stochastic resonance models are discussed.

Abstract

Suppose a two-dimensional dynamical system has a stable attractor that is surrounded by an unstable limit cycle. If the system is additively perturbed by white noise, the rate of escape through the limit cycle will fall off exponentially as the noise strength tends to zero. By analysing the associated Fokker-Planck equation we show that in general, the weak-noise escape rate is non-Arrhenius: it includes a factor that is periodic in the logarithm of the noise strength. The presence of this slowly oscillating factor is due to the nonequilibrium potential of the system being nondifferentiable at the limit cycle. We point out the implications for the weak-noise limit of stochastic resonance models.