Bifurcations in the Kuramoto model on graphs

Hayato Chiba, Georgi S. Medvedev, Matthew S. Mizuhara

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

In his classical work, Kuramoto analytically described the onset of synchronization in all-to-all coupled networks of phase oscillators with random intrinsic frequencies. Specifically, he identified a critical value of the coupling strength, at which the incoherent state loses stability and a gradual build-up of coherence begins. Recently, Kuramoto’s scenario was shown to hold for a large class of coupled systems on convergent families of deterministic and random graphs [Chiba and Medvedev, “The mean field analysis of the Kuramoto model on graphs. I. The mean field equation and the transition point formulas,” Discrete and Continuous Dynamical Systems - Series A (to be published); “The mean field analysis of the Kuramoto model on graphs. II. Asymptotic stability of the incoherent state, center manifold reduction, and bifurcations,” Discrete and Continuous Dynamical Systems - Series A (submitted).]. Guided by these results, in the present work, we study several model problems illustrating the link between network topology and synchronization in coupled dynamical systems. First, we identify several families of graphs, for which the transition to synchronization in the Kuramoto model starts at the same critical value of the coupling strength and proceeds in a similar manner. These examples include Erdős-Rényi random graphs, Paley graphs, complete bipartite graphs, and certain stochastic block graphs. These examples illustrate that some rather simple structural properties such as the volume of the graph may determine the onset of synchronization, while finer structural features may affect only higher order statistics of the transition to synchronization. Furthermore, we study the transition to synchronization in the Kuramoto model on power law and small-world random graphs. The former family of graphs endows the Kuramoto model with very good synchronizability: the synchronization threshold can be made arbitrarily low by varying the parameter of the power law degree distribution. For the Kuramoto model on small-world graphs, in addition to the transition to synchronization, we identify a new bifurcation leading to stable random twisted states. The examples analyzed in this work complement the results in Chiba and Medvedev, “The mean field analysis of the Kuramoto model on graphs. I. The mean field equation and the transition point formulas,” Discrete and Continuous Dynamical Systems - Series A (to be published); “The mean field analysis of the Kuramoto model on graphs. II. Asymptotic stability of the incoherent state, center manifold reduction, and bifurcations,” Discrete and Continuous Dynamical Systems - Series A (submitted).

Original languageEnglish
Article number073109
JournalChaos
Volume28
Issue number7
DOIs
Publication statusPublished - Jul 1 2018

Fingerprint

Kuramoto Model
Bifurcation
synchronism
Synchronization
dynamical systems
Graph in graph theory
Dynamical systems
Dynamical system
Mean Field
Random Graphs
Center Manifold Reduction
Mean Field Equation
Bifurcation (mathematics)
Series
Small World
transition points
Asymptotic stability
Asymptotic Stability
Critical value
Block graph

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

Cite this

Chiba, H., Medvedev, G. S., & Mizuhara, M. S. (2018). Bifurcations in the Kuramoto model on graphs. Chaos, 28(7), [073109]. https://doi.org/10.1063/1.5039609

Bifurcations in the Kuramoto model on graphs. / Chiba, Hayato; Medvedev, Georgi S.; Mizuhara, Matthew S.

In: Chaos, Vol. 28, No. 7, 073109, 01.07.2018.

Research output: Contribution to journalArticle

Chiba, H, Medvedev, GS & Mizuhara, MS 2018, 'Bifurcations in the Kuramoto model on graphs', Chaos, vol. 28, no. 7, 073109. https://doi.org/10.1063/1.5039609
Chiba H, Medvedev GS, Mizuhara MS. Bifurcations in the Kuramoto model on graphs. Chaos. 2018 Jul 1;28(7). 073109. https://doi.org/10.1063/1.5039609
Chiba, Hayato ; Medvedev, Georgi S. ; Mizuhara, Matthew S. / Bifurcations in the Kuramoto model on graphs. In: Chaos. 2018 ; Vol. 28, No. 7.
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