Synchronization Of Complex Networks With Mixed Time-varying Delays Based On Pulse Control

Complex network as a network form of complex system, is a frame representation which is highly abstracted from the complex system, is an essential abstraction of the interaction in complex systems, and it is prevalent in the nature and the human society. Synchronization is one of the most typical group behavior in complex networks,which has become an important research hotspot of network and attracted a lot of researchers' interest from different kinds of science to study it. And a large number of excellent research results are obtained. This paper will study the synchronization of the complex network with hybrid time-varying delay via impulsive control. The main work is as follows:First of all, we introduce the research background and significance of the exponential synchronization of complex network with hybrid time-varying delays via impulsive control. Summarizing and analyzing the shortcomings of the existing complex network model and related research achievements to introduce the method of no synchronization state in this paper. We also give several model of complex network and mathematical definition of complex network synchronization with the theory of complex network, which make the readers to understand the synchronization of complex network in the point view of mathematics.Secondly, we investigate the exponential synchronization of time-varying delays and non-delay coupled dynamical complex network via impulsive control. By utilizing the property of a special matrix “M” to construction the appropriate Lyapunov function and introducing the concept of "average impulsive interval" to study the network. According to the impulsive delay different inequality and Lyapunov Stability Theorem, we obtain some sufficient conditions which are derived to guarantee the global exponential synchronization for all initial values. Moreover, a few corollary are given for the special circumstances. These results we obtain which generalize some conclusion of the existing related research.Finally, we investigate the issue of exponential synchronization for a class of nonlinearly coupled complex networks with hybrid time-varying delays via impulsive control protocol, in the network all the time delay is time-varying.Different from the existing literature, we don't need to assume the synchronous goal to study the network synchronization, but use the difference of state variables between any two nodes in the network to discuss. By using Lyapunov Stability Theorem and "average impulsive interval" with the comparison theorem. Some sufficient conditions which is simple and easy to implement are derived to guarantee the global exponential synchronization for all initial values.And moreover, some illustrated numerical examples are given to verify the theoretical results we obtain, which show that the theoretical results are correct and has good stability.