Limit Cycles of Dynamic Systems under Random Perturbations with Rapid Switching and Slow Diffusion: A Multi-Scale Approach

This work is devoted to examining qualitative properties of dynamic systems, in particular, limit cycles of stochastic differential equations with both rapid switching and small diffusion. The systems are featured by multi-scale formulation, highlighted by the presence of two small parameters $ϵ$ and $\delta$. 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 ${\mu }^{ϵ,\delta }$, 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 ${\mu }^{ϵ,\delta }$ converges weakly to ${\mu }^0$ as $ϵ\to 0$ and $\delta \to 0$ under suitable conditions. Moreover, our results are applied to a stochastic predator-prey model together with numerical examples for demonstration.