Oscillatory decay of the survival probability of activated diffusion across a limit cycle

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Activated escape of a Brownian particle from the domain of attraction of a stable focus over a limit cycle exhibits non-Kramers behavior: it is non-Poissonian. When the attractor is moved closer to the boundary oscillations can be discerned in the survival probability. We show that these oscillations are due to complex-valued higher order eigenvalues of the Fokker-Planck operator, which we compute explicitly in the limit of small noise. We also show that in this limit the period of the oscillations is the winding number of the activated stochastic process. These peak probability oscillations are not related to stochastic resonance and should be detectable in planar dynamical systems with the topology described here.