The number of zeros of Abelian integrals for a perturbation of a hyper-elliptic Hamiltonian system with a nilpotent center and a cuspidal loop

In this paper we consider the number of isolated zeros of Abelian integrals associated to the perturbed system $\dot{x}=y,\ \dot{y}=-x^3(x-1)^2+\varepsilon (\alpha+\beta x+ \gamma x^3)y$, where $\varepsilon >0$ is small and $\alpha,\,\beta,\,\gamma \in \mathbb{R}$. The unperturbed system has a cuspidal loop and a nilpotent center. It is proved that three is the upper bound for the number of isolated zeros of Abelian integrals, and there exists some $\alpha,\,\beta$ and $\gamma$ such that the Abelian integrals could have three zeros which means three limit cycles could bifurcate from the nilpotent center and period annulus. The proof is based on a Chebyshev criterion for Abelian integrals, asymptotic behaviors of Abelian integrals and some techniques from polynomial algebra