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Article

Kuramoto Model with Delay: The Role of the Frequency Distribution

by
Vladimir V. Klinshov
1,2,3,4,* and
Alexander A. Zlobin
1,2,3
1
A. V. Gaponov-Grekhov Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul’yanov Street, 603950 Nizhny Novgorod, Russia
2
Faculty of Radiophysics of Nizhny Novgorod, Lobachevsky State University, 23 Prospekt Gagarina, 603022 Nizhny Novgorod, Russia
3
Leonhard Euler International Mathematical Institute, Saint Petersburg University, 7-9 Universitetskaya Embankment, 199034 St. Petersburg, Russia
4
National Research University Higher School of Economics, 25/12 Bol’shaya Pecherskaya Street, 603155 Nizhny Novgorod, Russia
*
Author to whom correspondence should be addressed.
Mathematics 2023, 11(10), 2325; https://doi.org/10.3390/math11102325
Submission received: 18 April 2023 / Revised: 7 May 2023 / Accepted: 10 May 2023 / Published: 16 May 2023
(This article belongs to the Special Issue Advances in Delay Differential Equations)

Abstract

The Kuramoto model is a classical model used for the describing of synchronization in populations of oscillatory units. In the present paper we study the Kuramoto model with delay with a focus on the distribution of the oscillators’ frequencies. We consider a series of rational distributions which allow us to reduce the population dynamics to a set of several delay differential equations. We use the bifurcation analysis of these equations to study the transition from the asynchronous to synchronous state. We demonstrate that the form of the frequency distribution may play a substantial role in synchronization. In particular, for Lorentzian distribution the delay prevents synchronization, while for other distributions the delay can facilitate synchronization.
Keywords: Kuramoto model; time delay; synchronization Kuramoto model; time delay; synchronization

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MDPI and ACS Style

Klinshov, V.V.; Zlobin, A.A. Kuramoto Model with Delay: The Role of the Frequency Distribution. Mathematics 2023, 11, 2325. https://doi.org/10.3390/math11102325

AMA Style

Klinshov VV, Zlobin AA. Kuramoto Model with Delay: The Role of the Frequency Distribution. Mathematics. 2023; 11(10):2325. https://doi.org/10.3390/math11102325

Chicago/Turabian Style

Klinshov, Vladimir V., and Alexander A. Zlobin. 2023. "Kuramoto Model with Delay: The Role of the Frequency Distribution" Mathematics 11, no. 10: 2325. https://doi.org/10.3390/math11102325

APA Style

Klinshov, V. V., & Zlobin, A. A. (2023). Kuramoto Model with Delay: The Role of the Frequency Distribution. Mathematics, 11(10), 2325. https://doi.org/10.3390/math11102325

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