Exact Results for the Kuramoto Model with a Bimodal Frequency Distribution

TL;DR

This paper provides an exact analytical characterization of the stability and bifurcation structure of the Kuramoto model with bimodal frequency distributions, revealing three distinct long-term dynamical states.

Contribution

It derives the complete stability diagram for the bimodal Lorentzian case using the Ott-Antonsen ansatz, advancing understanding of the model's global bifurcations.

Findings

Identification of three long-term states: incoherence, partial synchrony, standing wave.

Exact bifurcation boundaries between different dynamical regimes.

Extension of results to Gaussian bimodal distributions.

Abstract

We analyze a large system of globally coupled phase oscillators whose natural frequencies are bimodally distributed. The dynamics of this system has been the subject of long-standing interest. In 1984 Kuramoto proposed several conjectures about its behavior; ten years later, Crawford obtained the first analytical results by means of a local center manifold calculation. Nevertheless, many questions have remained open, especially about the possibility of global bifurcations. Here we derive the system's complete stability diagram for the special case where the bimodal distribution consists of two equally weighted Lorentzians. Using an ansatz recently discovered by Ott and Antonsen, we show that in this case the infinite-dimensional problem reduces exactly to a flow in four dimensions. Depending on the parameters and initial conditions, the long-term dynamics evolves to one of three states: incoherence, where all the oscillators are desynchronized; partial synchrony, where a macroscopic group of phase-locked oscillators coexists with a sea of desynchronized ones; and a standing wave state, where two counter-rotating groups of phase-locked oscillators emerge. Analytical results are presented for the bifurcation boundaries between these states. Similar results are also obtained for the case in which the bimodal distribution is given by the sum of two Gaussians.