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

Hayato Chiba, Georgi S. Medvedev

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)131-155
Number of pages25
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume39
Issue number1
DOIs
Publication statusPublished - Jan 2019

Fingerprint

Kuramoto Model
Mean Field Equation
Mean Field
Synchronization
Graph in graph theory
Synchrony
Coupled System
Critical value
Topology
Kernel Operator
Mean-field Limit
Topology Control
Small World
Random Graphs
Justification
Network Topology
Nearest Neighbor
Ensemble
Bifurcation
Linearly

All Science Journal Classification (ASJC) codes

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

The mean field analysis of the kuramoto model on graphs I. the mean field equation and transition point formulas. / Chiba, Hayato; Medvedev, Georgi S.

In: Discrete and Continuous Dynamical Systems- Series A, Vol. 39, No. 1, 01.2019, p. 131-155.

Research output: Contribution to journalArticle

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