Dynamics Analysis Of Impulsive Differential Systems With Pulse Phenomena

In this paper,we consider the stability property of the following functional differen-tial system(Ι) with state-dependent impulses where xt(s)=x(t+s),s∈[-τ,0].And pulse phenomena is allowed.As we know, the solutions of impulsive differential systems with state-dependent impulses may experience the pulse phenomena, namely the solutions may hit the same surface finite or infinite number of, times causing rhythmical beating. The presence of the phenomenon of 'beating' for such systems considerably complicates the investiga-tions. It is well known that impulsive differential systems provide mathematical models for many phenomena and processes in the field of natural sciences and technology. So that, in recent years, the study of impulsive differential systems has experienced a signif-icant development[1-5]. And there has also been various results for impulsive functional differential system[4-5]. Up to now, these results mostly focus on the functional differ-ential systems with fixed impulses and few takes the functional differential systems with state-dependent impulses into account, especially for the systems with pulse phenom-ena. However, the differential systems with pulse phenomena are more realistic and have wider application, so that there are still a lot of work to do in this field. In this paper, we focus on the research on dynamics analysis of functional differential systems with pulse phenomena. This paper is divided into three parts.In chapter one, we investigate the pulse phenomena of a class of impulsive differ-ential system. Some sufficient conditions that guarantee the absence and presence of pulse phenomena are obtained, without the boundedness requirement ofτκ. Our results generalize and improve several known results[1,6-9]. Then, an example is given to apply the theory.In chapter two, considering that the classical study of the stability of the nontrivial solution can not be instead by the study of the trivial solution, we study the stability of the nontrivial solution of system (Ⅰ). At first, we establish a comparison principle by comparing system (Ⅰ) with a ladder shaped differential systems. And then, by using the concepts of quasi-stability[10], several stability criteria for the nontrivial solution are established utilizing the known results[11], from which we can see the difficulties brought by the state-dependent impulses.In chapter three, we consider the stability of system (Ⅰ) mainly by the method of several Lyapunov functions containing partial components coupled with Razumikhin technique. In section 3, we investigate the stability properties in terms of two measures about the systems (Ⅰ), considering the (hoj,hj)-stability of the system. In section 4, we establish a comparison principle by comparing systems(Ⅰ) with an ordinary differential systems and using two Lyapunov functions containing partial components coupled with Razumikhin technique. Then, through constructing some special sets and utilizing Lya-punov functions with Razumikhin technique, we get some sufficient conditions for the stabilities of system (Ⅰ). It should be noticed that the pulse phenomena can be allowed. Some results in this chapter generalize and improve several known ones[12-13].