Title: A Multi-scale Approach to  Limit Cycles with Random Perturbations Involving Fast Switching and Small Diffusion

Abstract:
This talk is devoted to multi-scale stochastic systems. The motivation is to treat limit cycles under random perturbations involving fast  random switching and small diffusion, which are represented by the use of two small parameters. Associated with the underlying systems, there are averaged or limit systems. Suppose that for each pair of the parameters, the solution of the corresponding equation has an invariant probability measure $\mu^{\eps,\delta}$, and that the averaged equation has a limit cycle in which there is an averaged occupation measure $\mu^0$ for the averaged equation. Our main effort is to prove that $\mu^{\eps,\delta}$  converges weakly to $\mu^0$ as $\eps \to 0$ and $\delta \to 0$ under suitable conditions. We also examine  an application to a stochastic predator-prey model. Moreover, some numerical examples will also be reported.