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THE MEAN FIELD ANALYSIS OF THE KURAMOTO MODEL ON GRAPHS I. THE MEAN FIELD EQUATION AND TRANSITION POINT FORMULAS
Journal article   Open access   Peer reviewed

THE MEAN FIELD ANALYSIS OF THE KURAMOTO MODEL ON GRAPHS I. THE MEAN FIELD EQUATION AND TRANSITION POINT FORMULAS

Hayato Chiba, Georgi S. Medvedev and Institute of Mathematics for Industry, Kyushu University/JST PRESTO, Fukuoka, 819-0395, Japan
Discrete and continuous dynamical systems. Series A, v 39(1), pp 131-155
01 Jan 2019
DOI: https://doi.org/10.3934/dcds.2019006
url
https://doi.org/10.3934/dcds.2019006View
Published, Version of Record (VoR)Open Access (License Unspecified),  Open

Abstract

Mathematics Mathematics, Applied Physical Sciences Science & Technology
In his classical work on synchronization, Kuramoto derived the formula for the critical value of the coupling strength corresponding to the transition to synchrony in large ensembles of all-to-all coupled phase oscillators with randomly distributed intrinsic frequencies. We extend this result to a large class of coupled systems on convergent families of deterministic and random graphs. Specifically, we identify the critical values of the coupling strength (transition points), between which the incoherent state is linearly stable and is unstable otherwise. We show that the transition points depend on the largest positive or/and smallest negative eigenvalue(s) of the kernel operator defined by the graph limit. This reveals the precise mechanism, by which the network topology controls transition to synchrony in the Kuramoto model on graphs. To illustrate the analysis with concrete examples, we derive the transition point formula for the coupled systems on Erdos-Renyi, small-world, and k-nearest-neighbor families of graphs. As a result of independent interest, we provide a rigorous justification for the mean field limit for the Kuramoto model on graphs. The latter is used in the derivation of the transition point formulas. In the second part of this work [8], we study the bifurcation corresponding to the onset of synchronization in the Kuramoto model on convergent graph sequences.

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Mathematics
Mathematics, Applied
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