Switched systems with multiple invariant sets

•Under dwell time constraint, switched systems converge to a set in finite time.•Subsequently, trajectories remain within a larger invariant set.•Optimization of tuning parameters is a tradeoff between spatial/ temporal bounds.

This paper explores dwell time constraints on switched systems with multiple, possibly disparate invariant limit sets. We show that, under suitable conditions, trajectories globally converge to a superset of the limit sets and then remain in a second, larger superset. We show the effectiveness of the dwell-time conditions by using examples of switching limit cycles commonly found in robotic locomotion and flapping flight.