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We study the long-term qualitative behavior of randomly perturbed dynamicalsystems. More specifically, we look at limit cycles of stochastic differentialequations (SDE) with Markovian switching, in which the process switches atrandom times among different systems of SDEs, when the switching is fast andthe diffusion (white noise) term is small. The system is modeled by $$dX^{\epsilon,\delta}(t)=f(X^{\epsilon,\delta}(t),\alpha^\epsilon(t))dt+\sqrt{\delta}\sigma(X^{\epsilon,\delta}(t),\alpha^\epsilon(t))dW(t) , \ X^\epsilon(0)=x, $$ where $\alpha^\epsilon(t)$ isa finite state space Markov chain with irreducible generator $Q=(q_{ij})$. Therelative changing rates of the switching and the diffusion are highlighted bythe two small parameters $\epsilon$ and $\delta$. We associate to the systemthe averaged ODE \[ d\bar X(t)=\bar f(\bar X(t))dt, \ X(0)=x, \] where $\barf(\cdot)=\sum_{i=1}^{m_0}f(\cdot, i)\nu_i$ and $(\nu_1,\dots,\nu_{m_0})$ is theunique invariant probability measure of the Markov chain with generator $Q$.Suppose that for each pair $(\epsilon,\delta)$ of parameters, the process hasan invariant probability measure $\mu^{\epsilon,\delta}$, and that the averagedODE has a limit cycle in which there is an averaged occupation measure $\mu^0$for the averaged equation. We are able to prove that if $\bar f$ has finitelymany unstable or hyperbolic fixed points, then $\mu^{\epsilon,\delta}$converges weakly to $\mu^0$ as $\epsilon\to 0$ and $\delta \to 0$. Our resultsgeneralize to the setting of state-dependent switching \[\mathbb{P}\{\alpha^\epsilon(t+\Delta)=j~|~\alpha^\epsilon=i,X^{\epsilon,\delta}(s),\alpha^\epsilon(s), s\leqt\}=q_{ij}(X^{\epsilon,\delta}(t))\Delta+o(\Delta),~~ i\neq j \] as long as thegenerator $Q(\cdot)=(q_{ij}(\cdot))$ is bounded, Lipschitz, and irreducible forall $x\in\mathbb{R}^d$. We conclude our analysis by studying a predator-preymodel.

Dynamical systems with fast switching and slow diffusion: Hyperbolic equilibria and stable limit cycles