Bifurcation of Limit Cycles by Perturbing a Piecewise Linear Hamiltonian System

In this paper, we study limit cycle bifurcations for planar piecewise smooth near-Hamiltonian systems with nth-order polynomial perturbation. The linear differential two centers formed in ways, one is that a center-fold point at the origin, other origin and another unique center exists. We first explore expression of order Melnikov function. Then by using function method, give estimations number cycles bifurcating from period annulus. For latter case, simultaneous occurrence near both sides homoclinic loop partially addressed.