The exponential synchronization of Kuramoto oscillator networks in the presence of combined global and local cues

This work addresses the influences of global and local cues on the synchronization of Kuramoto oscillators coupled by nonlinear interactions. We first give a condition under which the network can be globally synchronized. This condition is more general than existing results and it is one of the first analytical results that ensure global synchronization of a general coupling structure when phase differences may be outside of the interval [-pi/2, pi/2]. Then we prove that the influence of global cues on the synchronization rate is always favorable, whereas the influence of local cues depends on the oscillators' relative phases (phases defined with reference to the phase of the global cue): if all the relative phases are within the interval [-pi/2, pi/2], the local cue either has no influence or increases the synchronization rate; otherwise the local cue may increase or decrease the synchronization rate. Finally, simulation results are given to illustrate the theoretical results.