Limit cycles near an eye-figure loop in some polynomial Liénard systems

In this paper, we study the number of limit cycles in the family dH−εω=0, where H=y22−∫0xg(u)du, ω=yf(x)dx, with g(x)=x(x2−1)(x2−14)2, and f(x) an even polynomial of degree 10. We will consider mainly the bifurcation of limit cycles near the eye-figure loop and the center of dH=0. Our investigation focuses on the lower bound of the maximal number of limit cycles for these systems. In particular, we show that the perturbed system can have at least 8 limit cycles when deg⁡(f(x))=10.