Abstract
We study complete synchronization of the complex dynamical networks described by linearly coupled ordinary differential equation systems (LCODEs). Here, the coupling is timevarying in both network structure and reaction dynamics. Inspired by our previous paper (Lu et al. (2007-2008)), the extended Hajnal diameter is introduced and used to measure the synchronization in a general differential system. Then we find that the Hajnal diameter of the linear system induced by the time-varying coupling matrix and the largest Lyapunov exponent of the synchronized system play the key roles in synchronization analysis of LCODEs with identity inner coupling matrix. As an application, we obtain a general sufficient condition guaranteeing directed time-varying graph to reach consensus. Example with numerical simulation is provided to show the effectiveness of the theoretical results.
Highlights
Complex networks have widely been used in theoretical analysis of complex systems, such as Internet, World Wide Web, communication networks, and social networks
For a time varying system denoted by DF, we can define its Hajnal diameter of the variational system (9) as follows: diam (DF, s0)
We present a theoretical framework for synchronization analysis of general coupled differential dynamical systems
Summary
Complex networks have widely been used in theoretical analysis of complex systems, such as Internet, World Wide Web, communication networks, and social networks. Linearly coupled ordinary differential equation systems (LCODEs) are a large class of dynamical systems with continuous time and state. A large number of papers have focused on the dynamical behaviors of coupled systems [3,4,5], especially the synchronizing characteristics. We consider synchronization of LCODEs with time-varying coupling.
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