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Dynamical systems with fast switching and slow diffusion: Hyperbolic equilibria and stable limit cycles. (arXiv:1902.01755v1 [math.PR])
来源于:arXiv
We study the long-term qualitative behavior of randomly perturbed dynamical
systems. More specifically, we look at limit cycles of certain 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 varying and the diffusion (white noise) term is slowly changing. The
system is modeled by $$ dX^{\varepsilon,\delta}(t)=f(X^{\varepsilon,\delta}(t),
\alpha^\varepsilon(t))dt+\sqrt{\delta}\sigma(X^{\varepsilon,\delta}(t),
\alpha^\varepsilon(t))dW(t) , \ X^\varepsilon(0)=x, $$ where
$\alpha^\varepsilon(t)$ is a finite state space, Markov chain with generator
$Q/\varepsilon=\big(q_{ij}/\varepsilon\big)_{m_0\times m_0}$ with $Q$ being
irreducible. The relative changing rates of the switching and the diffusion are
highlighted by the two small parameters $\varepsilon$ and $\delta$. We
associate to the system the averaged ordinary differential equation (ODE) \[
d\bar X(t)=\bar f(\bar X(t 查看全文>>