Stability diagram for the forced Kuramoto model

TL;DR

This paper provides a comprehensive bifurcation analysis of the forced Kuramoto model, revealing detailed stability diagrams and exact bifurcation points for a simplified case, enhancing understanding of synchronization phenomena.

Contribution

It offers the first complete bifurcation analysis of the forced Kuramoto model in a special case, deriving exact bifurcation locations and stability diagrams.

Findings

Exact locations of Hopf, saddle-node, and Takens-Bogdanov bifurcations.

Stability diagram similar to forced van der Pol oscillator.

Collapse of infinite-dimensional dynamics to a two-dimensional system.

Abstract

We analyze the periodically forced Kuramoto model. This system consists of an infinite population of phase oscillators with random intrinsic frequencies, global sinusoidal coupling, and external sinusoidal forcing. It represents an idealization of many phenomena in physics, chemistry and biology in which mutual synchronization competes with forced synchronization. In other words, the oscillators in the population try to synchronize with one another while also trying to lock onto an external drive. Previous work on the forced Kuramoto model uncovered two main types of attractors, called forced entrainment and mutual entrainment, but the details of the bifurcations between them were unclear. Here we present a complete bifurcation analysis of the model for a special case in which the infinite-dimensional dynamics collapse to a two-dimensional system. Exact results are obtained for the locations of Hopf, saddle-node, and Takens-Bogdanov bifurcations. The resulting stability diagram bears a striking resemblance to that for the weakly nonlinear forced van der Pol oscillator.