The Asymptotic Frequency of Stochastic Oscillators

We study stochastic perturbations of ODEs with stable limit cycles ---referred to as stochastic oscillators ---and investigate the response of the asymptotic (in time) frequency of oscillations to changing noise amplitude. Unlike previous studies, we do not restrict our attention to the small -noise limit, and we account for the fact that large deviation events may push the system out of its oscillatory regime. To do so, we consider stochastic oscillators conditioned on their remaining in an oscillatory regime for all time. This leads us to use the theory of quasi-ergodic measures, and to define quasi-asymptotic frequencies as conditional, long-time average frequencies. We show that quasi-asymptotic frequencies exist under minimal assumptions, though they may or may not be observable in practice. Our discussion recovers and expands upon previous results on stochastic oscillators in the literature. In particular, existing results imply that the asymptotic frequency of a stochastic oscillator depends quadratically on the noise amplitude. We describe scenarios where this prediction holds, though we also show that it is not true in general ---potentially, even for small noise.