Limit cycles in a Liénard system with a cusp and a nilpotent saddle of order 7

In this paper, we first give the topological classification of level curves for a special Liénard system. Then we study the number of limit cycles of some polynomial Liénard systems with a cuspidal loop surrounded by a loop that is connected (homoclinic) to a nilpotent saddle. We prove that H(5,6)⩾9,H(6,6)⩾10H(5,6)⩾9,H(6,6)⩾10 and H(7,6)⩾11H(7,6)⩾11, where H(m,n)H(m,n) is the maximal number of limit cycles in a Liénard system of type (m,n)(m,n).