Dynamics of the generalized Kuramoto model with nonlinear coupling: Bifurcation and stability

We systematically study dynamics of a generalized Kuramoto model of globally coupled phase oscillators. The coupling of modified model depends on the fraction of phase-locked oscillators via a power-law function of the Kuramoto order parameter r through an exponent α, such that α=1 corresponds to the standard Kuramoto model, α<1 strengthens the global coupling, and the global coupling is weakened if α>1. With a self-consistency approach, we demonstrate that bifurcation diagrams of synchronization for different values of α are thoroughly constructed from two parametric equations. In contrast to the case of α=1 with a typical second-order phase transition to synchronization, no phase transition to synchronization is predicted for α<1, as the onset of partial locking takes place once the coupling strength K>0. For α>1, we establish an abrupt desynchronization transition from the partially (fully) locked state to the incoherent state, whereas there is no counterpart of abrupt synchronization transition from incoherence to coherence due to that the incoherent state remains linearly neutrally stable for all K>0. For each case of α, by performing a standard linear stability analysis for the reduced system with Ott-Antonsen ansatz, we analytically derive the continuous and discrete spectra of both the incoherent state and the partially (fully) locked states. All our theoretical results are obtained in the thermodynamic limit, which have been well validated by extensive numerical simulations of the phase-model with a sufficiently large number of oscillators.