On the number of zeros of Abelian integral for some Liénard system of type (4, 3)

In this article, we study the Abelian integral M(h) corresponding to the following Liénard system,x˙=y,y˙=x3(x-1)+ε(a+bx+cx2+x3)y,where 0 < ε ≪ 1, a, b and c are real bounded parameters. Using the expansion of M(h) and a new algebraic criterion developed in Maeñosas and Villadelprat (2011) [6], we found that the lower and upper bounds of the maximal number of zeros of M are respectively 4 and 5. Hence, the above system can have 4 limit cycles and has at most 5 limit cycles bifurcating from the corresponding period annulus. The results obtained are new for this kind of Liénard system as we known.