  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.
