The Stability Analysis Of Impulsive Differential Systems With Pulse Phenomenon

It is known that the theory of impulsive differential systems provides a natural framework for mathematical modelling of many real-world phenomena , such as project, contral , communication , biology , economy and so on . There has been a significant development in the theory of impulsive differential equations since 90s, especially in the area where impulses are fixed . Even in the field of differential systems with state-dependent impulses , the study is conducted under the condition of every solution meeting each hypersurface exactly once. we know that pulse phenomena is a particular case of impulses, which alows trajectoties meet each hypersurface more than once. Still now , few results of impulsive differential systems with pulses have been obtained. Hence, there are a lot of work we need to do in this field. In this paper, we forcus on the research work on dynamics analysis of impulsive differential systems with pulse phenomenon. This paper is divided into three parts.In chapter one, we study the existence of pulse accumulation in second-order impulsive differential equations with variable times as follows:Firstly, we introduce the definition of the above system. Then give necessary and sufficient conditions to assume pulse accumulation with the method of analysis. Finally, two examples are given to show the effectiveness of the theorems.In chapter two, we study the stability in terms of two measures of impulsive differential svstems with variable times as follows: First in section 3, we establish a comparison principle by variationsl Lyapunov functions and comparing with an ordinary differential system, then apply it to investigate stability of the systems. Second in section 4. through constructing new function and sets, several interesting direct results which allow the Lyapunov function to increase or decrease along trajectories between succesive impulses and to swith back and forth between them at each impulse are established. It should be noticed that the pulse phenomena can be allowed in this chapter. To the best known of the author, the stability study of impulsive differential systems with pulse phenomenon is seldom seen.In chapter three, we obtain criterions for global stability and global exponential stability of the equilibrium point of Hopfield neural networks with state-dependent impulses by using Lyapunov functions and some obtained results in chapter two. Bisides, our results in this chapter generalize and improve several known results, which is beneficial to practical application.