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Abstract and Applied Analysis
Volume 2013, Article ID 134265, 15 pages
Research Article

Achieving Synchronization in Arrays of Coupled Differential Systems with Time-Varying Couplings

1School of Mathematical Sciences, Fudan University, Shanghai 200433, China
2Centre for Computational Systems Biology, Fudan University, Shanghai 200433, China
3Centre for Scientific Computing, The University of Warwick, Coventry CV4 7AL, UK
4School of Computer Science, Fudan University, Shanghai 200433, China

Received 10 April 2013; Accepted 5 June 2013

Academic Editor: Zidong Wang

Copyright © 2013 Xinlei Yi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.