Dwell time for switched systems with multiple equilibria on a finite time-interval

We describe the behavior of solutions of switched systems with multiple globally exponentially stable equilibria. We introduce an ideal attractor and show that the solutions of the switched system stay in any given $\varepsilon$-inflation of the ideal attractor if the frequency of switchings is slower than a suitable dwell time $T$. In addition, we give conditions to ensure that the $\varepsilon$-inflation is a global attractor. Finally, we investigate the effect of the increase of the number of switchings on the total time that the solutions need to go from one region to another.