The mean field analysis of the Kuramoto model on graphs II. Asymptotic stability of the incoherent state, center manifold reduction, and bifurcations

Hayato Chiba, Georgi S. Medvedev

Research output: Contribution to journalArticle

Abstract

In our previous work [3], we initiated a mathematical investigation of the onset of synchronization in the Kuramoto model (KM) of coupled phase oscillators on convergent graph sequences. There, we derived and rigorously justified the mean field limit for the KM on graphs. Using linear stability analysis, we identified the critical values of the coupling strength, at which the incoherent state looses stability, thus, determining the onset of synchronization in this model. In the present paper, we study the corresponding bifurcations. Specifically, we show that similar to the original KM with all-to-all coupling, the onset of synchronization in the KM on graphs is realized via a pitchfork bifurcation. The formula for the stable branch of the bifurcating equilibria involves the principal eigenvalue and the corresponding eigenfunctions of the kernel operator defined by the limit of the graph sequence used in the model. This establishes an explicit link between the network structure and the onset of synchronization in the KM on graphs. The results of this work are illustrated with the bifurcation analysis of the KM on Erdos-Rényi, small-world, as well as certain weighted graphs on a circle.

Original languageEnglish
Pages (from-to)3897-3921
Number of pages25
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume39
Issue number7
DOIs
Publication statusPublished - 2019 Jul
Externally publishedYes

Keywords

  • Bifurcation
  • Center manifold reduction
  • Graph limit
  • Infinite-dimensional dynamical system
  • Mean field limit
  • Random graph
  • Synchronization

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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