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

TL;DR

This paper investigates the behavior of limit cycles in stochastic differential equations with rapid switching and slow diffusion, demonstrating convergence of invariant measures in a multi-scale setting and applying findings to predator-prey models.

Contribution

It establishes the weak convergence of invariant measures for multi-scale stochastic systems with rapid switching and slow diffusion, providing new insights into their qualitative behavior.

Findings

Invariant measures converge to the averaged system as parameters tend to zero

Results are validated through numerical examples in predator-prey models

Provides a theoretical framework for analyzing multi-scale stochastic systems

Abstract

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 $\epsilon$ 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 $\mu^{\epsilon,\delta}$, 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 $\mu^{\epsilon,\delta}$ converges weakly to $\mu^0$ as $\epsilon \to 0$ and $\delta \to 0$ under suitable conditions. Moreover, our results are applied to a stochastic predator-prey model together with numerical examples for demonstration.