A new approach to bifurcations in the Kuramoto model
type: Item
instance of: scholarly article
title: A new approach to bifurcations in the Kuramoto model
zbMATH Open document ID: 1477.37059
author: Timothy Ferguson
published in: Journal of Mathematical Analysis and Applications
publication date: 2021-06-14T00:00:00Z
review text: The author considers the Kuramoto model [\textit{Y. Kuramoto}, Chemical oscillations, waves, and turbulence. Berlin etc.: Springer-Verlag (1984; Zbl 0558.76051)] for the dynamics of networks of coupled oscillators. This model has found widespread application in the study of synchronisation, as reported in [\textit{S. H. Strogatz}, Physica D 143, No. 1--4, 1--20 (2000; Zbl 0983.34022)]. Here, bifurcations involving phase-locked solutions -- those for which all oscillators rotate at a common frequency with fixed angular differences -- on a ring network are investigated. The author derives criteria for a branch of stable phase-locked solutions to collide with a branch of \(1\)-saddles as the coupling parameter is varied, and illustrates those in specific examples. Finally, he studies the bistability phenomenon, i.e., the existence of two distinct stable phase-locked solutions, as well as higher-order bifurcations. His analysis is largely based on a novel vector flow which yields a local parametrisation of the governing equations close to the bifurcation regime.
Mathematics Subject Classification ID: 37G10
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 34C15
zbMATH DE Number: 7358698
zbMATH Keywords: Kuramoto model
zbMATH Keywords: synchronisation
zbMATH Keywords: phase-locking
zbMATH Keywords: bifurcation
zbMATH Keywords: bistability
reviewed by: Nikola Popović
MaRDI profile type: Publication
full work available at URL: j.jmaa.2021.125205
OpenAlex ID: W3153690054
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DOI: 10.1016/J.JMAA.2021.125205
description: scientific article; zbMATH DE number 7358698