In this paper the asymptotic expansion of first-order Melnikov function of a heteroclinic loop connecting a cusp and a nilpotent saddle both of order one for a planar near-Hamiltonian system are given. Next, we consider the bifurcation of limit cycles of a class of hyper-elliptic Liénard system with this kind of heteroclinic loop. It is shown that this system can undergo Poincarè bifurcation from which at most three limit cycles for small positive ε can emerge in the plane. Also using this

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... totic expansion it was shown that there exist parameter values for which three limit cycles exist close to this loop.