Stability Analysis Of Impulsive Functional Differential Systems And Their Application In Complex Networks

Research in the field of modern science and technology and engineering has greatly developed with each passing day. The mathematical model of a large number of practical problems is often come down to the nonlinear differential equation. Impulsive functional differential system, which is a kind of discrete system, not only describes the phenomena of transient process, but also reflects the influence of the moment in the past or the fu-ture time on the current state of the system. Therefore, impulsive functional differential system has been widely applied in neural network, optical control, population dynamics, biotechnology, economics etc [22,34-37]. The study on the properties of such systems, includes basic theory, geometry theory, stability theory, vibration theory etc. The dy-namic system has become a hot research topic for many mathematicians who have made some good researches.Relative to the sense of Lyapunov stability theory, in practice, the concept of prac-tical stability can reflect the essence of the research process, so the research about the property of the practical stability of differential system is more and more focused by mathematics workers, and some good results have been achieved etc. However, there are many problems worth to explore. In this paper, from a practical perspective, we mainly studies the stability of impulsive functional differential system. The solution of the system with given initial estimate area in advance and then estimate area, makes many unstable systems in Lyapunov sense achieve stability in the actual sense. At the same time, under the pulse disturbance or impulse control conditions, we get global expo-nential stability of the results with given convergence rate, overcoming the situation that we can’t control convergence rate. And for a special kind of pulse neural network system, we get its specific global exponential stability with given convergence rate which is the result of more practical significance than before. This paper is focused on the dynamics analysis of impulsive functional differential system and gives affirmative answer to some of the issues. This paper is divided into three chapters. In chapter one, we study the impulsive functional differential system with finite or infinite delays as follows By using Lyapunov functions and Razumikhin technique, the sufficient conditions of prac-tical stability of impulsive differential functional system with finite delay in terms of two measurements and uniformly asymptotically practical stability of impulsive differential functional system with infinite delay in terms of two measurements are obtained. Under these conditions, some systems which are unstable in Lyapunov cense can be practical stable.In chapter two, we study the impulsive functional differential system as follows and impulsive neural networks as follows Globally exponential stability of system (1.1.2) with given convergence rate has been established, overcoming the difficulties that the convergence rate cannot be controlled, and for a specific pulse neural network system without time-delay (1.1.3), by using the properties of positive definite matrix, negative definite matrix to get the global exponen-tial stability with the given convergence rate α. In the case of constant coefficient, it is easy to use Matlab to verify.In chapter three, we mainly study practical stability of the impulsive Hopfeld Neural Networks mode as follows Using the results in the first two chapters, we establish conditions on the practical prop-erties of the system (1.1.4).