Bifurcations of synchronized solutions in a continuum limit of the Kuramoto model with two-mode interaction depending on two graphs

TL;DR

This paper analyzes bifurcations in a continuum limit of the Kuramoto model with two-mode interactions on different graph structures, revealing new stable solutions and contrasting behaviors with classical models.

Contribution

It introduces a bifurcation analysis of the continuum Kuramoto model with two-mode interactions on various graphs using center manifold reduction, highlighting novel solution families.

Findings

Bifurcations lead to stable two-parameter families of sinusoidal solutions.

The continuum limit exhibits bifurcation behavior different from the classical Kuramoto model.

Numerical simulations confirm theoretical predictions for zero natural frequency.

Abstract

We study bifurcations of the completely synchronized state in a continuum limit (CL) for the Kuramoto model (KM) of identical oscillators with two-mode interaction depending on two graphs. Here one of the graphs is uniform but may be deterministic dense, random dense or random sparse, and the other is a deterministic finite nearest neighbor. We use the center manifold reduction technique, which is a standard one in dynamical systems, and prove that the CL suffers bifurcations at which the one-parameter family of completely synchronized state becomes unstable and a stable two-parameter family of $\ell$-humped sinusoidal shape stationary solutions ($\ell\ge 2$) appears. This contrasts the author's recent result on the classical KM for which bifurcation behavior in its CL is very different from ones in the KM and difficult to explain by standard techniques in dynamical systems such as the center manifold reduction. Moreover, similar bifurcation behavior is shown to occur in the KM, based on the previous fundamental results. The occurrence of such bifurcations were suggested by numerical simulations for the deterministic graphs in a previous study. We also demonstrate our theoretical results by numerical simulations for the KM with the zero natural frequency.