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.

一组耦合的仓本振荡器是物理，生物和工程网络中振动现象协调研究的主要应用模型。为了与先前的研究相一致，并使分析结果与进一步的实际模型相符，本文对仓本振子的同步进行了研究，并利用收缩特性介绍了频率同步和相位凝聚力的必要和充分条件。 。本文的新颖性在于：（I）考虑耦合拓扑中具有非均匀性的异构二阶模型；（II）在耦合函数中应用非零且非均匀的相移；（III）我们引入了一种基于Lyapunov的新稳定性分析技术。我们证明了网络的异质性和耦合函数的相移是网络同步的关键因素。根据网络图的理论特性和振荡器的参数，给出了同步条件。分析结果的研究表明，振荡器的相移和异质性的增加将使同步条件复杂化。数值模拟证实了主要理论结果的有效性。分析结果的研究表明，振荡器的相移和异质性的增加将使同步条件复杂化。数值模拟证实了主要理论结果的有效性。分析结果的研究表明，振荡器的相移和异质性的增加将使同步条件复杂化。数值模拟证实了主要理论结果的有效性。