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

Hayato Chiba, Georgi S. Medvedev

研究成果: Contribution to journalArticle査読

9 被引用数 (Scopus)

抄録

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.

本文言語英語
ページ(範囲)131-155
ページ数25
ジャーナルDiscrete and Continuous Dynamical Systems- Series A
39
1
DOI
出版ステータス出版済み - 1 2019

All Science Journal Classification (ASJC) codes

  • 分析
  • 離散数学と組合せ数学
  • 応用数学

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