A Hopf bifurcation in the Kuramoto-Daido model

TL;DR

This paper analyzes a Hopf bifurcation in the Kuramoto-Daido model, demonstrating the emergence of a periodic two-cluster state through spectral theory and center manifold reduction.

Contribution

It introduces a novel analysis of bifurcation behavior in the Kuramoto-Daido model using advanced spectral and reduction techniques.

Findings

Hopf bifurcation occurs as coupling strength increases

Existence of a periodic two-cluster state is proven

Dynamical system on a four-dimensional center manifold is derived

Abstract

A Hopf bifurcation in the Kuramoto-Daido model is investigated based on the generalized spectral theory and the center manifold reduction for a certain class of frequency distributions. The dynamical system of the order parameter on a four-dimensional center manifold is derived. It is shown that the dynamical system undergoes a Hopf bifurcation as the coupling strength increases, which proves the existence of a periodic two-cluster state of oscillators.