Limit Cycles Of A Class Of Cubic Liénard Equations

In this paper, a class of polynomial Lienard system is studied. Lienard system has a wide range of application in many practical fields, such as in mechanical oscillation, chemical reaction, radio electronics circuit, the population dynamics, nerve stimulation and non-linear mechanics. It can be used to describe the heart beat, the circuit circulation, the assignments of conveyor belt and the working condition of communication equipment, and it is playing an ever increasingly greater role. Lienard system contains many specific equations with practical background, so it is of certain practical significance to research Lienard system. In addition, many poly-nomial systems can be transformed into Lienard system through proper transformation, then the research results from Lienard system can be used to study the existence, non-existence and uniqueness of polynomial systems' limit cycles.According to the number and type of singularities in the system (1), the paper mainly considers the following four cases:(ⅰ)b2≠0, (ⅱ) b2= 0,b1= 0,b3≠0, (ⅲ)b2= 0,b1b3> 0, (ⅳ) b2= 0,b1b3< 0. In this paper, Filippov transformation and Zhang Zhifen's theorem are used as the main methods. On the basis of Zhang's theorem, the paper makes a generalization of the theorem.Through the research on the system(1),some conditions of the existence,non-existence and uniqueness of limit cycles are obtained.And,one example is given to illustrate that the system can have three limit cycles.Our main results are the following.Conclusion 1 The system(1)has at most one limit cycle surrounding the origin ifα1α3<0 or b2=0.Conclusion 2 For b2=0,the system(2)has1)no any closed orbits if b1=0,b3<0 or bl=0,b3>0,α1α3≥0 or b1<0,b3<0 or b1>0,b3>0,α1α3≥0 or b1>0,b3<0,(?)>(?)or b1>0,b3<0,α1α3≥0.2)a unique limit cycle if bl≥0,b3>0 andα1α3<0.3) three limit cycles surrounding a singular point if b1=-1,α1=-(?)-ε-2δ-γ,α2=2(?)+δ+2γor-2(?)-δ-2γ,α3=-(?)-γ, where 0<ε《δ《γ《1.