Tuning limit cycles with a noise: Survival and collapse.

Abstract

We consider a general class of limit cycle oscillators driven by an additive Gaussian white noise. Based on the separation of timescales, we construct the equationof motion for slow dynamics after appropriate averaging over the fast motion. The equationfor slow motion whose coefficients are modified by noise characteristics is solved to obtain the analytic solution in the long time limit. We show that with increase of noise strength, the loop area of the limit cycle decreases until a critical value is reached, beyond which the limit cycle collapses. We determine the noise threshold from the condition for removal of secular divergence of the perturbation series and work out two explicit examples of Van der Pol and Duffing-Van der Pol oscillators for corroboration between the theory and numerics.