Exponential synchronization of Kuramoto oscillators with time delayed coupling

TL;DR

This paper establishes conditions under which non-identical Kuramoto oscillators with time delays synchronize exponentially, extending previous results and providing explicit bounds based on initial configurations.

Contribution

It generalizes existing synchronization estimates to include time delays and provides explicit bounds on coupling strength and delay for exponential synchronization.

Findings

Explicit lower bound on coupling strength derived

Upper bound on time delay established

Synchronization conditions depend on initial phase configurations

Abstract

We discuss the asymptotic frequency synchronization for the non-identical Kuramoto oscillators with time delayed interactions. We provide explicit lower bound on the coupling strength and upper bound on time delay in terms of initial configurations ensuring exponential synchronization. This generalizes not only the frequency synchronization estimate by Choi et al. [Physica D, 241, (2012), 735-754] for the non-identical Kuramoto oscillators without time delays but also improves previous result by Schmidt et al. [Automatica, 48, (2012), 3008-3017] in the case of homogeneous time delays where the initial phase diameter is assumed to be less than ${\pi}/2$. The proof relies on a Lyapunov functional approach.