Low-dimensional behavior of generalized Kuramoto model

We study the global bifurcation of a generalization of the Kuramoto model in the fully connected network in which the connections are weighted by the frequency of the oscillators. By driving the low dimensional manifold of this infinite-dimensional dynamical system, we obtain bifurcation boundaries for different types of transitions to the synchronized state. Using this analytic framework, we obtain the characteristic flow field of the system for each dynamical region in parameter space. To check the effect of nonzero-centered frequency distribution, we consider bimodal Lorentzian distribution as an example. In this case, the system shows three types of transitions to the synchronized state, depending on the parameters of the frequency distribution: (1) a two-step transition with Bellerophon state, (2) a continuous transition, as in the classical Kuramoto model, and (3) a first-order, explosive, transition with hysteresis.