Communications in Mathematical Sciences

Volume 14 (2016)

Number 4

Emergence of phase-locked states for the Kuramoto model in a large coupling regime

Pages: 1073 – 1091

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n4.a10

Authors

Seung-Yeal Ha (Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, South Korea)

Hwa Kil Kim (School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea)

Sang Woo Ryoo (Department of Mathematical Sciences, Seoul National University, Seoul, South Korea)

Abstract

We study the emergence of phase-locked states to the finite-dimensional Kuramoto model from generic initial configurations which are not phase-locked states in a large coupling regime. In the literature of physics and engineering, it has often been argued that complete synchronization may occur for a generic initial configuration in a large coupling regime. Such arguments are generally based on the results of numerical simulations. Unfortunately, this plausible scenario has not been completely verified by rigorous mathematical arguments, although there are several partial results available for a restricted class of initial configurations. In this paper, we provide a sufficient framework for complete synchronization from a generic initial configuration in a large coupling regime. Our analysis depends on the gradient flow structure of the Kuramoto model and the uniform boundedness of the phase configuration.

Keywords

complete synchronization, Kuramoto model, order parameter, phase-locked states

2010 Mathematics Subject Classification

15B48, 92D25

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Published 6 April 2016