Robust Asymptotic Stability of Desynchronization in Impulse-Coupled Oscillators

TL;DR

This paper proves that desynchronization in a network of impulse-coupled oscillators is asymptotically stable and robust to perturbations, using hybrid system stability analysis and Lyapunov methods.

Contribution

It introduces a hybrid system model for impulse-coupled oscillators and demonstrates the asymptotic stability and robustness of desynchronization.

Findings

Desynchronization is asymptotically stable in the model.

Desynchronization remains robust under various perturbations.

Numerical simulations confirm theoretical results.

Abstract

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.