Communications in Mathematical Sciences

Volume 19 (2021)

Number 5

Exponential synchronization of Kuramoto oscillators with time delayed coupling

Pages: 1429 – 1445



Young-Pil Choi (Department of Mathematics, Yonsei University, Seoul, South Korea)

Cristina Pignotti (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila, Italy)


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 the time delay in terms of initial configurations ensuring exponential synchronization. This generalizes not only the frequency synchronization estimate by Choi et al. [Phys. D, 241(7):735–754, 2012] for the non-identical Kuramoto oscillators without time delays but also improves previous result by Schmidt et al. [Automatica, 48(12):3008–3017, 2012] 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.


Kuramoto model, frequency synchronization, time delay, Lyapunov functional approach

2010 Mathematics Subject Classification

34C15, 34D05, 92D25

Y.-P. Choi was supported by POSCO Science Fellowship of the POSCO TJ Park Foundation, and by Yonsei University Research Fund of 2019-22-0212.

C. Pignotti was supported by GNAMPA-INdAM and RIA-UNIVAQ.

Received 6 May 2020

Accepted 13 January 2021

Published 11 November 2021