【摘要】 复杂网络同步是复杂系统和复杂网络的前沿研究方向之一,已经取得很大的进展.但是对于节点以耦合矩阵左特征向量加权平均态、孤立节点的解与网络的同步态之间具有什么关系,什么是网络的同步态和同步轨等基本问题仍然缺乏深入的研究,弄清楚这些问题对于复杂网络同步的理解和应用具有重要的意义.本文采用数学分析方法证明,如果网络同步,则加权平均态xˉ=N j=1ξj xj可以定义为同步态,一般来说,xˉ在正极限集的意义下,也就是孤立节点方程s˙(t)=f(s(t))的解.因此在实际应用中,把孤立节点方程的解s(t)与加权平均态xˉ不加区别地对待是合理的.同步态是不依赖于初始条件的通解,而同步轨是依赖于初始条件的特解.对于混沌节点的网络,同步态应该理解为吸引子,而不是某一条轨道.最后,本文还提供一些实例加以说明,并指出一些尚待解决的问题.更多还原

【Abstract】 Synchronization is one of the frontier researches in complex systems and complex networks, which has been fruitfully exploited. However, further research is still necessary for some fundamental questions, such as the definitions of synchronous state and synchronous orbit of a dynamical network, relationships among the weighted mean state, the solution of the individual system and the synchronous state of the entire network. It is of great importance to address these issues so as to contribute to an integrated understanding and practical applications of synchronization in complex networks. In this paper, mathematical analysis is used to demonstrate that if a network synchronizes, the synchronous state can be defined as the weighted mean state xˉ =N j=1 ξj xj, which is the solution of the isolated system s˙(t) = f(s(t)) in the sense of the positive limit set. Therefore, there is no diference between the solution of the individual system s(t) and the weighted mean state xˉ in practice. Compared to the synchronous state which is a general solution independent of initial conditions, the synchronous orbit is a special solution related to initial conditions. As for networks coupled with chaotic systems, the synchronous state should be viewed as attractors, instead of a particular orbit. Finally, numerical simulations are provided to illustrate the efectiveness of our theoretical results, and some problems needed to be further studied are also included.更多还原