# Convergence and Synchronization in Networks of Piecewise-Smooth Systems via Distributed Discontinuous Coupling

## TL;DR

This paper introduces a method to achieve global asymptotic synchronization in networks of piecewise-smooth systems by adding a distributed discontinuous coupling, providing analytical thresholds and a new connectivity measure called minimum density.

## Contribution

It proposes a novel distributed discontinuous coupling protocol for synchronization in piecewise-smooth networks, with analytical convergence conditions and the concept of minimum density as a connectivity measure.

## Key findings

- Analytical thresholds for coupling gains are derived.
- Distributed discontinuous coupling effectively enforces synchronization.
- The minimum density measure predicts network synchronization success.

## Abstract

Complex networks are a successful framework to describe collective behaviour in many applications, but a notable gap remains in the current literature, that of proving asymptotic convergence in networks of piecewise-smooth systems. Indeed, a wide variety of physical systems display discontinuous dynamics that change abruptly, including dry friction mechanical oscillators, electrical power converters, and biological neurons. In this paper, we study how to enforce global asymptotic state-synchronization in these networks. Specifically, we propose the addition of a distributed discontinuous coupling action to the commonly used diffusive coupling protocol. Moreover, we provide analytical estimates of the thresholds on the coupling gains required for convergence, and highlight the importance of a new connectivity measure, which we named minimum density. The theoretical results are illustrated by a set of representative examples.