Multiple-parameter bifurcation analysis in a Kuramoto model with time delay and distributed shear

TL;DR

This paper analyzes how time delay and distributed shear influence the stability and bifurcation behavior of the Kuramoto model, revealing rich dynamics including synchronization loss, stability switches, and multiple coherent states.

Contribution

It provides a comprehensive bifurcation analysis of the Kuramoto model with time delay and distributed shear, deriving stability boundaries and characterizing complex dynamical phenomena.

Findings

Shear distribution width and time delay can suppress synchronization.

Time delay induces multiple coexisting coherent states.

Numerical simulations confirm theoretical bifurcation analysis.

Abstract

In this paper, time delay effect and distributed shear are considered in the Kuramoto model. On the Ott-Antonsen's manifold, through analyzing the associated characteristic equation of the reduced functional differential equation, the stability boundary of the incoherent state is derived in multiple-parameter space. Moreover, very rich dynamical behavior such as stability switches inducing synchronization switches can occur in this equation. With the loss of stability, Hopf bifurcating coherent states arise, and the criticality of Hopf bifurcations is determined by applying the normal form theory and the center manifold theorem. On one hand, theoretical analysis indicates that the width of shear distribution and time delay can both eliminate the synchronization then lead the Kuramoto model to incoherence. On the other, time delay can induce several coexisting coherent states. Finally, some numerical simulations are given to support the obtained results where several bifurcation diagrams are drawn, and the effect of time delay and shear is discussed.