The Synchronization Stability Analysis Of A Kind Of Complex Dynamical Networks

In recent years, scholars pay more and more attentions to complex networks, which already have been widely used in science and technology fields. In fact, we already live in a world which is surrounded by complex networks. The power grid, transportation network, internet, neural networks, social networks, etc., can be seen as complex networks. A cause for great concern issues of complex works is its synchronization problems. This paper introduces the synchronization of complex dynamical networks, including chaotic synchronization, and mainly study the synchronization stability of complex dynamical networks with delay. By constructing Lyapunov-Krasovskii functional, according to the Lyapunov stability theorem, this paper proposed two new determination methods for synchronization stability in the presence of continuous, and proved the feasibility of them in theory. The results of numerical simulation further indicate the actual feasibility. The functional structure in previous literatures often contains tree terms, excluding integral term, integral term, quadratic integral term. The functional structure in this paper used integral term instead of quadratic integral term. By researching the new Lyapunov-Krasovskii functional, this paper deduced the third-order linear matrix inequality. The problem of synchronization stability with delay changed into the problem of linear matrix inequality. The linear matrix inequality has simple structure, and can be solved easily by matlab. In most of the existing literature, positive definite symmetric matrices used to construct Lyapunov-Krasovskii functional always are constant matrices. This article introduced time variable into one of the constant matrices, analyzing the new Lyapunov-Krasovskii functional, and then the problem of synchronization stability with delay transformed into the problem of differential equation with positive definite solution. This method is for the case that delay is known. In addition, for the discrete case, this paper also proposed a new determination method about synchronization stability, which is for the delay-independent, and also verified the feasibility of this approach in theory.