The Estimate Of The Amplitude Of Limit Cycles Of Symmetric Liénard Systems

In this paper,symmetric Lienard system x = y-F(x),y =-g(x)(i.e.F(x)and g(x)are odd functions)is mainly studied.It is well known that under some hypotheses,this system has a unique limit cycle.Under some additional conditions,we develop a method to give both the upper bound and lower bound of the amplitude,which is the maximal value of the x-coordinate,of the unique limit cycle.As an application,we consider van der Pol equation x = y ＝ μ(x3/3-x),y =-x,where μ>0.From some classical conclusions,we know that van der Pol equation has a unique limit cycle,denote by A(μ)the amplitude of its limit cycle,then for any μ,we show that A(μ)0<(?)≈2.0976 and for μ= 1,2,we show that A(μ)>2.Both the upper bound and the lower bound improve the existing ones.