Qualitative Research Of Two Kinds Of Nonlinear Systems

The research of limit cycle is one of the most important and difficult problem in the differential equation qualitative theory.This problem received extensive attention of the mathematician,physicists and thechnology scientists.There have many resear ch results of quadratic polynomial system and cubic polynomial system,but there have relatively little results of higher order polynomial system.This paper mainly studies following two higher order polynomial system: This is two types of Lienard polynomial system.As is known to all, the Lienard system has a wide range of applications in many practical fields.In areas such as the mechanical shock,radio electronic circuit,chemical reaction,population dynamics, nonlinear mechanics and nerve stimulation.At the same time,it can also be used to describe the circuit loop,the heart beats,the working condition of conveyor belt and communication equipment and so on.Along with the further research,the Lienard system will further play a important role.All of these show that the research of the Lienard system has practical significance.In the aspect of theory,many polynomial system can be converted to the Lienard system through certain transformation.Then we could use the existing research results of the Lienard system to research these polynomial system.This is another important role of the Lienard system.In this paper,we use the Filippov transform,the Zhang Zhifen theorem and the Zhang Zhifen theorem which is improved by Jin Huatao to research the above two high order polynimial system.According to the number and the type of the singularity, the above four polynomial system can be divided into the following several ways to discuss:(1)a4=0,b4≠0,b2=b3=0;(2)a4＝0, b4≠0, b2＝O；(3)a4=0,b4≠0,b3=0; (4)a4=0,b4≠0,b1=b2=0:(5)a4=0,b4≠0,b1=b3=0:(6)a4=0,b4≠0,b1=b2=b3=0(7)a4≠0,b4=0,b2=b3=0:(8)a4≠0,b4=0,b1=b2=0:(9)a4≠0,b4=0,b2=0:(10)a4≠0,b4=0,b1=b3=0:(11)a4=0,b4≠0,a1=a3=b2=b4=0. As to five polynimial system, it can be divided into the following six ways to discuss:(2)a1=a2=a4=0,a3<0,a5>0,|a3|≥a5;(3)a2=a3=a4=0,a1<0,a5>0;(4)a3=a4=0,a1<0,a22<0,a5>0,|a1|≥5a,a2≥a1;(5)a2=a3=0,a1<0,a4<0,a5>0,|a1|≥a5,a4≥a1;(6)a1=a4=0,a2<0,a3<0,a5>0,|a3|≥a5,a2≥a3.Through the case discussion,we can obtain related conditions of the existence, nonexistence and uniqueness of limit cycle of the four polynomial system.Also obta-in some conclusions of uniqueness of limit cycle of the five polynomial system.