Exact bound on the number of limit cycles arising from a periodic annulus bounded by a symmetric heteroclinic loop

In this paper, the bound on the number of limit cycles by Poincare bifurcation in a small perturbation of some seventh-degree Hamiltonian system is concerned. The lower and upper bounds on the number of limit cycles have been obtained in two previous works, however, the sharp bound is still unknown. We will employ some new techniques to determine which is the exact bound between $3$ and $4$. The asymptotic expansions are used to determine the four vertexes of a tetrahedron, and the sharp bound can be reached when the parameters belong to this tetrahedron.