DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2011.16.393 DYNAMICAL SYSTEMS SERIES B Volume 16, Number 1, July 2011 pp. 393–408 LOCAL AND GLOBAL EXPONENTIAL SYNCHRONIZATION OF COMPLEX DELAYED DYNAMICAL NETWORKS WITH GENERAL TOPOLOGY Jin-Liang Wang National Key Laboratory of Science and Technology on Holistic Control School of Automation Science and Electrical Engineering Beihang University, Beijing 100191, China Zhi-Chun Yang College of Mathematics Science, Chongqing Normal University Chongqing 400047, China Tingwen Huang Department of Mathematics and Science Texas A&M University at Qatar PO Box 23874, Doha, Qatar Mingqing Xiao Department of Mathematics, Southern Illinois University Carbondale, IL 62901, USA (Communicated by David Yang Gao) Abstract. In this paper, we consider a generalized complex network possess- ing general topology, in which the coupling may be nonlinear, time-varying, nonsymmetric and the elements of each node have different time-varying de- lays. Some criteria on local and global exponential synchronization are derived in form of linear matrix inequalities (LMIs) for the complex network by con- structing suitable Lyapunov functionals. Our results show that the obtained sufficient conditions are less conservative than ones in previous publications. Finally, two numerical examples and their simulation results are given to illus- trate the effectiveness of the derived results. 1. Introduction. In the past ten years, there have been many researchers study- ing the topology and dynamical behavior of complex networks across many fields of science and engineering, such as power grids, communication networks, Internet, World Wide Web, metabolic systems, food webs and so on [1, 2]. In particular, the synchronization is one of the most significant and interesting dynamical prop- erties of the complex networks. Many interesting results on synchronization have been derived, e.g., see also [3]-[25]. In [3], a general complex delayed dynamical 2000 Mathematics Subject Classification. Primary: 37N99, 93D05; Secondary: 93A30. Key words and phrases. Complex networks, time-varying delays, exponential synchronization. This work was supported in part by National Natural Science Foundation of China under Grants 10971240, 6100404 & 61074057, in part by Natural Science Foundation of Chongqing under Grant CSTC2008BB2364, in part by Foundation of Science and Technology project of Chongqing Education Commission under Grant KJ080806, in part by Fundamental Research Funds for the Central Universities of China under Grant YWF-10-01-A19, in part by NSF 1021203 of the U.S.. 393