On the Relationship between the Synchronous State and the Solution of an Isolated Node in a Complex Network
Chen Jua · Acta Automatica Sinica · 2013
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.