The Stability Analysis Of Certain First Order Impulsive Delay Differential Systems With Pulse Phenomenon

In this paper, we mainly study stability properties of the following impulsive delay differential system: and where pulse phenomena is allowed in (Ⅰ) and (Ⅰ).Impulsive differential systems is a new branch of mathematics rising from the early 1980s. The stability analysis of it is an important research,and is also an focus and dif-ficult problem of non-liner dynamic systems research in the word. There are so many great and widespread realistic background in natural science to research impulsive de-lay differential equation.Many problems in mathematic model,such as electric analysis in physical aspect,ecology science,mental disorder in biological science,population science and economics can be summarized to study impulsive delay differential equation,so There are much important practical value to research them.The research of it gets many experts and scholars’s attention,and some very good results are put forward .The solutions of impulsive differential systems with stated-dependent impulses may experience the pulse phenomena,namely the solutions may hit the same surface finite or infinite number of times casing rhythmical beating.The presence of the phenomenon of "beating" for such systems considerably complicates the investigations.Still now,few results of impulsive differential equations with pulse phenomena have been obtained.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 phe-nomena.Hence,it is necessary to research the system.In the nature,many thing’s variance not only relied on conditions then,but also re-lied on conditions for its past time,namely there is a lag between reaction and the reason which caused it,and often with impulsive phenomenon followed.All these mathematical model can attach to the delay impulsive differential systems,which is more richer and realistic than the impulsive differential systems,which is more richer and realistic than the impulsive differential systems.There are many achievement for study in the recent ten years,and the 3/2 stability method is very useful.The research of impulsive integro-differential system has aroused expert’s interest and attention,and has made important progress in recent years,but the research of delay impulsive integro-differential sys-tems is quite scarce,and there are also multiply works to do.This paper focuses on the pulsation of the first-order impulsive delay differential system stability analysis, using the 3/2 method to analysis the stability of the zero solu-tion of the system is given to be uniformly stable as well as asymptotic stability of the specific conditions,the full text is divided into two parts.In chapter one I mainly research one kind of the stability of certain first order constant coefficient impulsive delay differential systems with pulse phenomenon.In first,I introduce relation prior knowledge about impulsive delay differential equation,then,over-come the effects of pulse moments are not fixed,reasonable assumption for the system is put forward.For the curve and pulse collision free moments are not fixed,points of discussion,use 3/2 method,I get a sufficient condition of the solution and applications.In chapter two I mainly research the stability of the form of another kind of certain first order variable coefficient impulsive delay differential systems with pulse phenomenon more specific.On the basis of the first chapter,given a similar conclusion.First,I introduce relevant theoretical knowledge,overcome the effects of pulse moments, consider difference t,similarly,I get the sufficient condition of the solution.