Synchronization in Networks with Strongly Delayed Couplings

TL;DR

This paper analyzes the stability of synchronization in networks with strong delays, deriving conditions and critical coupling strengths that depend on network structure and dynamics, with implications for large and random networks.

Contribution

It provides new strict conditions for synchronization stability in delayed networks and introduces a critical coupling strength concept that depends on network and dynamical properties.

Findings

Synchronization stability depends on coupling strength and delay.

Critical coupling strength can be independent of delay for equilibria.

Synchronization interval is maximized near network connectivity threshold.

Abstract

We investigate the stability of synchronization in networks of dynamical systems with strongly delayed connections. We obtain strict conditions for synchronization of periodic and equilibrium solutions. In particular, we show the existence of a critical coupling strength $\kappa_{c}$, depending only on the network structure, isolated dynamics and coupling function, such that for large delay and coupling strength $\kappa<\kappa_{c}$, the network possesses stable synchronization. The critical coupling $\kappa_{c}$ can be chosen independently of the delay for the case of equilibria, while for the periodic solution, $\kappa_{c}$ depends essentially on the delay and vanishes as the delay increases. We observe that, for random networks, the synchronization interval is maximal when the network is close to the connectivity threshold. We also derive scaling of the coupling parameter that allows for a synchronization of large networks for different network topologies.