Results on the asymptotic stability properties of desynchronization in impulse-coupled oscillators

The property of desynchronization in impulse-coupled oscillators is studied. Each impulsive oscillator is modeled as a hybrid system with a single timer state that self-resets to zero when it reaches a threshold, at which point any other impulsive oscillator adjusts their timers following a common law. This law dictates the reaction to an external reset. In this setting, desynchronization is considered as timers having equal separation among each other and between successive resets. We show that, for the considered model, desynchronization is an (almost global) asymptotic stability property, which, due to the regularity properties of the hybrid systems, is robust to small perturbations. To establish this result, we recast desynchronization as a set stabilization problem and employ Lyapunov stability tools for hybrid systems. The results are illustrated in examples and simulations.