Synchronization transition of the second-order Kuramoto model on lattices

TL;DR

This study investigates the synchronization transition of the second-order Kuramoto model on large 2D and 3D lattices, revealing hybrid phase transitions and critical behaviors influenced by initial conditions and dimensionality.

Contribution

It provides the first detailed numerical analysis of the second-order Kuramoto model's phase transition on lattices, identifying hybrid transitions and critical exponents.

Findings

Discontinuous transition for Gaussian frequencies in 2D and 3D.

Evidence of hybrid phase transitions with crossover behavior.

Critical decay of frequency spread as t^{-d/2} in aligned initial states.

Abstract

The second-order Kuramoto equation describes synchronization of coupled oscillators with inertia, which occur in power grids for example. Contrary to the first-order Kuramoto equation it's synchronization transition behavior is much less known. In case of Gaussian self-frequencies it is discontinuous, in contrast to the continuous transition for the first-order Kuramoto equation. Here we investigate this transition on large 2d and 3d lattices and provide numerical evidence of hybrid phase transitions, that the oscillator phases $\theta_i$, exhibit a crossover, while the frequency spread a real phase transition in 3d. Thus a lower critical dimension $d_l^O=2$ is expected for the frequencies and $d_l^R=4$ for the phases like in the massless case. We provide numerical estimates for the critical exponents, finding that the frequency spread decays as $\sim t^{-d/2}$ in case of aligned initial state of the phases in agreement with the linear approximation. However in 3d, in the case of initially random distribution of $\theta_i$, we find a faster decay, characterized by $\sim t^{-1.8(1)}$ as the consequence of enhanced nonlinearities which appear by the random phase fluctuations.