An upper bound for the amplitude of limit cycles in Liénard systems with symmetry

It is well known that the Liénard system x˙=y−F(x), y˙=−g(x) with symmetry (i.e. F(x)F(x) and g(x)g(x) are odd functions) has a unique limit cycle under some hypotheses. In this paper we will show that the unique limit cycle locates in the strip region |x|0x⁎>0 is uniquely and directly determined by ∫0x⁎F(x)g(x)dx=0. In other words, an explicit upper bound x⁎x⁎ is given for the amplitude (i.e. the maximal value of the x  -coordinate) of the unique limit cycle. As a simple application we obtain a uniform estimate A(μ)<5≐2.2361 for each μ>0μ>0, where A(μ)A(μ) denotes the amplitude of the unique limit cycle in the Liénard system x˙=y−μ(x3/3−x), y˙=−x for the van der Pol equation x¨+μ(x2−1)x˙+x=0. The upper bound 5 improves the existing ones.