Pulse Phenomena And Stability Analysis For Impulsive Differential Equations With Variable Times

There has been a significant development in the theory of impulsive differential equations in the past 20 years,especially in the area where impulses are fixed[17,18] and few takes the impulsive differential equations with variable times impulses into account. The main differ-ence between equations with fixed impulses and equations with variable times impulses is pulse phenomena.We know that pulse phenomena is a particular case of impulses,which allows trajec-toties meet each hypersurface more than once,and it is very common in the actual problems and gets more people’s attention.Pulse phenomena presents additional difficulties in the investigation of properties of solu-tions of such systems. Still now,few results of impulsive differential equations with pulse phe-nomena have been obtained.Pulse accumulation is a special situation that when time tends to be a limited time,the solution of the system experiences infinite pulses and tend to be a little eventu-ally.In fact, the impulsive differential equations without pulse phenomena is only a kind of ideal model.The actual problems are complex systems that allowed the happen of pulse phenomena.Up to now,the results about pulse phenomena in second-order impulsive differential equations with variable times are really few,and most of them are about the first-order impulsive differential equations with variable times.Hence,there are a lot of work to do in this field.In this paper we focus on the research on impulsive differential equations with pulse phenomena.This paper is divided into two parts.In chapter one,in the conditions of x0<x* and x0> x*, we investigate the existence of pulse accumulation directly in a class of impulsive differential equation with variable times as follows:Firstly,we weaken the conditions of Tk(x) to unbounded,and give some sufficient conditions to assume pulse accumulation and the solution meets the given surface firstly with the method of analysis.And then, we consider the complex situation that before the pulse accumulation ap-peared in the given surface,the solution may meet the other surface first.In order to get the ex-pected effect,we need to consider the proper conditions of every elements.In chapter two,we study the presence of pulse phenomena and existence of pulse accumu-lation in second-order impulsive differential equations with variable times as follows: andFirstly,we give some sufficient conditions which can guarantee the presence of pulse phe-nomena of the above equations without considering the boundness ofτk(x) andτ(x).And then in the conditions of x0≥x*,y0≥y*,we investigate the existence of pulse accumulation in second-order impulsive differential equation with variable times.On the base of others work,we mainly changed some conditions ofτ(x), f(t, x,x). It should be pointed that,(x*, y*)is the pulse accumu-lation.Our results generalize and improve several known results.And the last,an example is given to apply the theory.