Analysis Of Limit Cycles On Two Differential Systems

Limit cycle problem is an important problem in the qualitative theoretical research of differential equations and it is also part of the 16th century problem raised by Hilbert.In this paper,the existence,stability,uniqueness and quantity of limit cycles of two kinds of differential systems are studied based on the qualitative and stability theories of differential equations.In the paper we firstly study a class of cubic polynomial Lienard systems.We consider the existence and stability of limit cycles for this class of system.Based on the method of solutions the condition a1a3 0 can be divided into three small conditions.By using Filippov transformation,Zhangzhifen theorem and Bendixson-Dulac theorem,we get some conclusions.When b2?0,b3<0,b1>0,a1=0,a3>0,there is at most one limit cycle,and it's stable if exists.When b2?0,b3<0,b1>0,a1#0,a3=0,there is no limit cycle existed.The empty of limit cycle properties of cubic polynomial Lienard system under coefficient condition of a1a3?0 is added.Secondly,in this paper a class of piecewise linear system added a high-order term is studied.We make use of the system which can directly solve the solution trajectory of system and the situation that the trajectory from the same initial point will meet at y-axis again.From Calculating coordinate of the meeting point,which means conditions for the existence of limit cycle,we obtain that piecewise system exists at most one limit cycle,which enriched the research of limit cycle joined a high-order term.Finally,in this paper we combine the two systems studied before and considers the limit cycle properties of piecewise Lienard systems.Based on the existing conclusions from the above two systems and the singularities analysis,by using the method of Bendixson-Dulac theorem,we finally conclude that when the piecewise Lienard system has five singularities and meets the corresponding conditions,there are two limit cycles,in which the inner limit cycle is stable and the outer limit cycle is unstable.The relationship between the two systems in this paper is improved.The conclusion is an important supplement to the limit cycle of piecewise Lienard system which is seldom studied.