TY - JOUR

T1 - The mean field analysis of the kuramoto model on graphs I. the mean field equation and transition point formulas

AU - Chiba, Hayato

AU - Medvedev, Georgi S.

N1 - Funding Information:
74A25, 92B25. Key words and phrases. Mean field limit, synchronization, graph limit, random graph. The second author was partially supported by NSF DMS grant 1715161. ∗ Corresponding author: Georgi S. Medvedev.
Publisher Copyright:
© Copyright 2018 American Institute of Mathematical Sciences.

PY - 2019/1

Y1 - 2019/1

N2 - 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 Erdös-Rényi, 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.

AB - 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 Erdös-Rényi, 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.

KW - Graph limit

KW - Mean field limit

KW - Random graph

KW - Synchronization

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U2 - 10.3934/dcds.2019006

DO - 10.3934/dcds.2019006

M3 - Article

AN - SCOPUS:85056140106

SN - 1078-0947

VL - 39

SP - 131

EP - 155

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

IS - 1

ER -