The Kuramoto Model: The Stability Conditions in the Presence of Phase Shift

摘要

A set of coupled Kuramoto oscillators is the main applied model for harmonization study of oscillating phenomena in physical, biological and engineering networks. In line with the previous studies and to bring the analytical results into conformity with further realistic models, in present paper the synchronization of Kuramoto oscillators has been investigated and the necessary and sufficient conditions for the frequency synchronization and phase cohesiveness have been introduced using the contraction property. The novelty of this paper lies in the following: (I) we consider the heterogeneous second-order model with non-uniformity in coupling topology; (II) we apply a non-zero and non-uniform phase shift in coupling function; (III) we introduce a new Lyapunov-based stability analysis technique. We demonstrate how the heterogeneity in network and the phase shift in coupling function are the key factors in network synchronization. The synchronization conditions are presented on the basis of network graph-theoretical characteristics and the oscillators’ parameters. Investigation of the analytical results reveals that an increase in the phase shift and heterogeneity of oscillators will complicate the synchronization conditions. The validity of the main theoretical results has been confirmed through the numerical simulations.