Towards a general theory for coupling functions allowing persistent synchronization

TL;DR

This paper develops a general theoretical framework for coupling functions that enable persistent and stable synchronization in complex networks of coupled dynamical systems, even under perturbations.

Contribution

It introduces a class of coupling functions that guarantee uniform and persistent synchronization in complex networks, extending previous results to more general settings.

Findings

Stable synchronization is achieved under the proposed coupling functions.

Synchronization persists despite perturbations to node dynamics.

Numerical examples validate the theoretical results.

Abstract

We study synchronisation properties of networks of coupled dynamical systems with interaction akin to diffusion. We assume that the isolated node dynamics possesses a forward invariant set on which it has a bounded Jacobian, then we characterise a class of coupling functions that allows for uniformly stable synchronisation in connected complex networks --- in the sense that there is an open neighbourhood of the initial conditions that is uniformly attracted towards synchronisation. Moreover, this stable synchronisation persists under perturbations to non-identical node dynamics. We illustrate the theory with numerical examples and conclude with a discussion on embedding these results in a more general framework of spectral dichotomies.