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Dynamical systems with fast switching and slow diffusion: Hyperbolic equilibria and stable limit cycles
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 where is
a finite state space Markov chain with irreducible generator . The
relative changing rates of the switching and the diffusion are highlighted by
the two small parameters and . We associate to the system
the averaged ODE where and is the
unique invariant probability measure of the Markov chain with generator .
Suppose that for each pair of parameters, the process has
an invariant probability measure , and that the averaged
ODE has a limit cycle in which there is an averaged occupation measure
for the averaged equation. We are able to prove that if has finitely
many unstable or hyperbolic fixed points, then
converges weakly to as and . Our results
generalize to the setting of state-dependent switching as long as the
generator is bounded, Lipschitz, and irreducible for
all . We conclude our analysis by studying a predator-prey
model.Comment: 40 page
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