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

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    We study the long-term qualitative behavior of randomly perturbed dynamical systems. More specifically, we look at limit cycles of stochastic differential equations (SDE) with Markovian switching, in which the process switches at random times among different systems of SDEs, when the switching is fast and the diffusion (white noise) term is small. The system is modeled by dXϵ,δ(t)=f(Xϵ,δ(t),αϵ(t))dt+δσ(Xϵ,δ(t),αϵ(t))dW(t), Xϵ(0)=x, 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 αϵ(t)\alpha^\epsilon(t) is a finite state space Markov chain with irreducible generator Q=(qij)Q=(q_{ij}). The relative changing rates of the switching and the diffusion are highlighted by the two small parameters ϵ\epsilon and δ\delta. We associate to the system the averaged ODE dXˉ(t)=fˉ(Xˉ(t))dt, X(0)=x, d\bar X(t)=\bar f(\bar X(t))dt, \ X(0)=x, where fˉ()=i=1m0f(,i)νi\bar f(\cdot)=\sum_{i=1}^{m_0}f(\cdot, i)\nu_i and (ν1,,νm0)(\nu_1,\dots,\nu_{m_0}) is the unique invariant probability measure of the Markov chain with generator QQ. Suppose that for each pair (ϵ,δ)(\epsilon,\delta) of parameters, the process has an invariant probability measure μϵ,δ\mu^{\epsilon,\delta}, and that the averaged ODE has a limit cycle in which there is an averaged occupation measure μ0\mu^0 for the averaged equation. We are able to prove that if fˉ\bar f has finitely many unstable or hyperbolic fixed points, then μϵ,δ\mu^{\epsilon,\delta} converges weakly to μ0\mu^0 as ϵ0\epsilon\to 0 and δ0\delta \to 0. Our results generalize to the setting of state-dependent switching P{αϵ(t+Δ)=j  αϵ=i,Xϵ,δ(s),αϵ(s),st}=qij(Xϵ,δ(t))Δ+o(Δ),  ij \mathbb{P}\{\alpha^\epsilon(t+\Delta)=j~|~\alpha^\epsilon=i, X^{\epsilon,\delta}(s),\alpha^\epsilon(s), s\leq t\}=q_{ij}(X^{\epsilon,\delta}(t))\Delta+o(\Delta),~~ i\neq j as long as the generator Q()=(qij())Q(\cdot)=(q_{ij}(\cdot)) is bounded, Lipschitz, and irreducible for all xRdx\in\mathbb{R}^d. We conclude our analysis by studying a predator-prey model.Comment: 40 page
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