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arXiv · 2211.15497

Synchronization transition of the second-order Kuramoto model on lattices

Géza Ódor·Shengfeng Deng
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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.

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Géza Ódor, Shengfeng Deng. 2022-11-28. Synchronization transition of the second-order Kuramoto model on lattices. https://doi.org/10.3390/e25010164

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