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The property of desynchronization in an all-to-all network of homogeneous impulse-coupled oscillators is studied. Each impulse-coupled oscillator is modeled as a hybrid system with a single timer state that self-resets to zero when it reaches a threshold, at which event all other impulse-coupled oscillators adjust their timers following a common reset law. In this setting, desynchronization is considered as each impulse-coupled oscillator's timer having equal separation between successive resets. We show that, for the considered model, desynchronization is an asymptotically stable property. For this purpose, we recast desynchronization as a set stabilization problem and employ Lyapunov stability tools for hybrid systems. Furthermore, several perturbations are considered showing that desynchronization is a robust property. Perturbations on both the continuous and discrete dynamics are considered. Numerical results are presented to illustrate the main contributions.

The problem of synchronization of multiple linear time-invariant systems connected over a network with asynchronous and intermittently available communication events is studied. To solve this problem, we propose a controller with hybrid dynamics, namely, the controller utilizes information transmitted to it during discrete communication events and exhibits continuous dynamics between such events. Due to the additional continuous and discrete dynamics inherent to the interconnected networked systems and communication structure, we use a hybrid systems framework to model and analyze the closed-loop system. The problem of synchronization is then recast as a set stabilization problem and, by employing Lyapunov stability tools for hybrid systems, sufficient conditions for asymptotic stability of the synchronization set are provided. Furthermore, we show that the property of synchronization is robust to perturbations. Numerical examples illustrating the main results are included.

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This paper introduces incremental stability notions for a class of hybrid dynamical systems given in terms of differential equations and difference equations with state constraints. The specific class of hybrid systems considered are those that do not have consecutive jumps nor Zeno behavior. The notion of incremental asymptotic stability is used to describe the behavior of the distance between every pair of solutions to the system having stable behavior (incremental stability) and approaching zero asymptotically (incremental attractivity). A version of this notion that is uniform (in hybrid time) with respect to initial conditions is also introduced. These notions prioritize flow time and are illustrated in examples. Basic properties of the class of systems are considered and those implied by the new notions are revealed. An equivalence characterization of the uniform notion is provided in terms of a KL-function. Moreover, sufficient and necessary conditions under which asymptotic stability implies the new incremental notions are provided. We consider the case when the original hybrid system has an asymptotically stable compact set and also the case when an auxiliary hybrid system, which has twice the dimension of the original system, has a diagonal-like set asymptotically stable.