Limit cycle bifurcations by perturbing a kind of quadratic Hamiltonian system having a homoclinic loop

In this paper, we investigate the number of limit cycles bifurcating from a kind of quadratic Hamiltonian system having a homoclinic loop under perturbations of piecewise smooth polynomials with degree $ n $. It is proved that at most $ 10n+[\frac {n-1}2]+7 (n\geq 1) $ limit cycles bifurcating from the periodic annulus if the first order Melnikov function is not identically zero, and it is also proved that there exist piecewise smooth polynomials with degree $ n $ such that the corresponding system has at least $ n+[\frac{n+1}{2}] $ limit cycles.