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

TL;DR

This paper analyzes the bifurcation and stability of the incoherent state in the Kuramoto model on graphs, establishing explicit links between network structure and synchronization onset through mean field analysis and eigenfunction analysis.

Contribution

It extends mean field analysis of the Kuramoto model to graph sequences, deriving bifurcation formulas involving eigenvalues and eigenfunctions, and linking network structure to synchronization.

Findings

Synchronization onset occurs via a pitchfork bifurcation.

Explicit formulas relate bifurcation to eigenvalues of the graph kernel.

Results demonstrated on Erdős-Rényi, small-world, and weighted circle graphs.

Abstract

In our previous work [Chiba, Medvedev, arXiv:1612.06493], 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 bifurcation. 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 Erd\H{o}s-R{\' e}nyi, small-world, as well as certain weighted graphs on a circle.