On the reachable set of inflated singularly perturbed differential equations

Singularly perturbed differential equations with slow and fast subsystems are under consideration. It is well-known that the invariant probability measures of the unperturbed fast subsystems produce a limiting system for the slow subsystem. Unfortunately, the limiting system is non-smooth in general and might be too big since it also generates trajectories that are not related to the singularly perturbed system. However, if the flows produced by the unperturbed fast subsystems are chain transitive an inflation of the singularly perturbed system can be used in order to approximate all trajectories of the limiting system. Moreover, it turns out that the reachable sets of the limiting system are contained in the reachable sets of the slightly inflated singularly perturbed system.