Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp

In this paper we study the limit cycle bifurcation of a piecewise smooth Hamiltonian system. By using the Melnikov function of piecewise smooth near-Hamiltonian systems, we obtain that at most \begin{document} $12n+7$ \end{document} limit cycles can bifurcate from the period annulus up to the first order in \begin{document} $\varepsilon$ \end{document} .