The Asymptotic Frequency of Stochastic Oscillators

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