Synchronization Transition of the Second-Order Kuramoto Model on Lattices

The second-order Kuramoto equation describes the synchronization of coupled oscillators with inertia, which occur, for example, in power grids. On the contrary to the first-order Kuramoto equation, its synchronization transition behavior is significantly less known. In the case of Gaussian self-frequencies, it is discontinuous, in contrast to the continuous transition for the first-order Kuramoto equation. Herein, we investigate this transition on large 2D and 3D lattices and provide numerical evidence of hybrid phase transitions, whereby the oscillator phases [Formula: see text] exhibit a crossover, while the frequency is spread over a real phase transition in 3D. Thus, a lower critical dimension [Formula: see text] is expected for the frequencies and [Formula: see text] for phases such as that in the massless case. We provide numerical estimates for the critical exponents, finding that the frequency spread decays as [Formula: see text] in the case of an aligned initial state of the phases in agreement with the linear approximation. In 3D, however, in the case of the initially random distribution of [Formula: see text] , we find a faster decay, characterized by [Formula: see text] as the consequence of enhanced nonlinearities which appear by the random phase fluctuations.