Persistence of periodic solutions from discontinuous planar piecewise linear Hamiltonian differential systems with three zones

In this paper, we study the number of limit cycles that can bifurcate from a period annulus in discontinuous planar piecewise linear Hamiltonian differential system with three zones separated by two parallel straight lines. More precisely, we consider the case where the period annulus, bounded by a heteroclinic orbit or homoclinic loop, is obtained from a real center of the central subsystem, i.e. the system defined between the two parallel lines, and two real saddles of the others subsystems. Denoting by H(n) the number of limit cycles that can bifurcate from this period annulus by polynomial perturbations of degree n, we prove that if the period annulus is bounded by a heteroclinic orbit then H(1) >= 2, H(2) >= 3 and H(3) >= 5. Now, if the period annulus is bounded by a homoclinic loop then H(1) >= 3, H(2) >= 4 and H(3) >= 7. For this, we study the number of zeros of a Melnikov function for piecewise Hamiltonian system.