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