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

Abstract

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This work addresses the influences of global and local cues on the synchronization of Kuramotooscillators 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 [-π/2, π/2].Then we prove that the influence of global cues onthe synchronization rate is always favorable, whereas the influence of local cues depends on theoscillators' relative phases (phases defined with reference to the phase of the global cue): if all therelative phases are within the interval [-π/2, π/2], the local cue either has no influence or increases thesynchronization rate; otherwise the local cue may increase or decrease the synchronization rate. Finally, simulation results are given to illustrate the theoretical results.