The estimate of the amplitude of limit cycles of symmetric Liénard systems

Symmetric Liénard system x˙=y−F(x),y˙=−g(x) (i.e. F(x) and g(x) are odd functions) is studied. It is well known that under some hypotheses, this system has a unique limit cycle. 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. Denote by A(μ) the amplitude of its unique limit cycle, then for any μ, we show that A(μ)<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.