Bifurcations in the time-delayed Kuramoto model of coupled oscillators: Exact results

TL;DR

This paper derives exact results for how time delay influences the stability and bifurcation behavior in the Kuramoto model of coupled oscillators, using unstable manifold expansion and the Ott-Antonsen ansatz.

Contribution

It provides a rigorous analysis of delay effects on bifurcation in the Kuramoto model, connecting unstable manifold theory with the Ott-Antonsen reduction.

Findings

Delay significantly alters bifurcation characteristics.

The Ott-Antonsen ansatz aligns with unstable manifold results near bifurcation.

Delay can dramatically change bifurcation in bimodal frequency distributions.

Abstract

In the context of the Kuramoto model of coupled oscillators with distributed natural frequencies interacting through a time-delayed mean-field, we derive as a function of the delay exact results for the stability boundary between the incoherent and the synchronized state and the nature in which the latter bifurcates from the former at the critical point. Our results are based on an unstable manifold expansion in the vicinity of the bifurcation, which we apply to both the kinetic equation for the single-oscillator distribution function in the case of a generic frequency distribution and the corresponding Ott-Antonsen(OA)-reduced dynamics in the special case of a Lorentzian distribution. Besides elucidating the effects of delay on the nature of bifurcation, we show that the approach due to Ott and Antonsen, although an ansatz, gives an amplitude dynamics of the unstable modes close to the bifurcation that remarkably coincides with the one derived from the kinetic equation. Further more, quite interestingly and remarkably, we show that close to the bifurcation, the unstable manifold derived from the kinetic equation has the same form as the OA manifold, implying thereby that the OA-ansatz form follows also as a result of the unstable manifold expansion. We illustrate our results by showing how delay can affect dramatically the bifurcation of a bimodal distribution.