theory for hybrid systems, we prove that the desynchronization set is asymptotically stable by defining a Lyapunov-like function as the distance between the state and (an inflated version of) the desynchronization set. In our context, asymptotic stability of the desynchronization set implies that the distance between the state and the desynchronization set converges to zero as the amount of time and the number of jumps get large. Using the proposed Lyapunov-like function and invoking an invariance principle, the basin of attraction is characterized and shown to be the entire state space minus a set of measure zero, which turns out to actually be an exact estimate of the basin of attraction. Furthermore, also exploiting the availability of a Lyapunov-like function, we analytically characterize the time for the solutions to reach a neighborhood of the desynchronization set. In particular, this characterization provides key insight for the design of algorithms used in applications in which desynchronization is crucial, such as wireless digital networks and sensor networks.

The asymptotic stability property of the desynchronization configuration is shown to be robust to several types of perturbations. The perturbations studied here include a generic perturbation in the form of an inflation of the dynamics of 