  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.
