Synchronization And Passivity Analysis Of Complex Delayed Dynamical Networks

Complex networks widely exist in the world, from Internet to World Wide Web, from communication networks to social networks,from cellular networks to metabolic networks etc,and thus attract increasing attention of researchers from different fields.It is well known that one of the most significant and interesting dynamical properties of complex network is synchronization motion of its dynamical elements.Synchronization may be useful in many situations, such as confidential communications,the emergence of language and its development, and the coordination of organizational management and its effcient operation. However, synchronization is harmful in some situations, for instantce, information congestion in Internet or communications network,periodic routing messages from different routers, the increasing of the transmission control protocol windows, etc. Therefore, study of synchronization in complex networks is very important both in theory and application. On the other hand, the passivity of control systems depicts the relationship between input and output,is also an important concept of system theory. However, to our knowledge, few authors have considered the problem on the passivity of complex networks. Therefore, it is interesting to study the passivity of complex dynamical networks with time-varying delays.In this paper, we study the synchronization and passivity of complex dynamical networks associated with time delays effects.In the first part of our paper, we analyze the state-synchronization and output-synchronization properties of complex delayed dynamical networks.Firstly, we formulate a complex dynamical network model possessing general topology, in which the coupling may be nonlinear, time-varying and nonsymmetric and the elements of each node have different time-varying delays. Some criteria on local and global exponential synchronization are derived for dynamical networks by constructing some suitable Lyapunov functional and using linear matrix inequalities (LMI) approach. Furthermore, we discuss the output-synchronization of complex delayed dynamical networks using the property of Laplacian matrix of connected graph. In the second part of our paper, we investigate the input and output passivity of complex delayed dynamical networks, respectively. Firstly,we introduce the concepts of the input and output passivity for a class of delayed dynamical networks. Some input and output passivity criteria are derived for complex dynamical networks using Lyapunov functionals together with linear matrix inequality (LMI) approach.