# Article

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Keywords:
group synchronization; coupled harmonic oscillators; directed topology; acyclic partition
Summary:
This paper considers group synchronization issue of diffusively directed coupled harmonic oscillators for two cases with nonidentical and identical agent dynamics. For the case of coupled nonidentical harmonic oscillators with positive coupling, it is demonstrated that distributed group synchronization can always be achieved under two kinds of network structures, i. e., the strongly connected graph and the acyclic partition topology with a directed spanning tree. It is interesting to find that the group synchronization states under acyclic partition are some periodic orbits with the same frequency and are simply related with the initial values of certain group regardless of ones of the other groups. For the case of coupled identical harmonic oscillators with positive and negative coupling, some generic algebraic criteria on group synchronization with both local continuous and instantaneous interaction are established respectively. In particular, an explicit expression of group synchronization states in terms of initial values of the agents can be obtained by the property of acyclic partition topology, and so it is very convenient to yield the desired group synchronization in practical application. Finally, numerical examples illustrate and visualize the effectiveness and feasibility of theoretical results.
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