Four crossing limit cycles of a family of discontinuous piecewise linear systems with three zones separated by two parallel straight lines

In this paper, we study the maximum number of limit cycles that can exhibit a planar discontinuous piecewise differential system separated by two parallel straight lines and formed by two arbitrary linear differential systems with isolated singularity in the lines of discontinuity and a linear Hamiltonian saddle. More precisely, we prove that when the piecewise differential systems are of type boundary focus-Hamiltonian linear saddle-boundary focus, then this class of systems has at most four crossing limit cycles. But when the piecewise differential system is of type boundary focus-Hamiltonian linear saddle-boundary center, we show that it can have at most three limit cycles, and when the piecewise differential system is of type boundary center-Hamiltonian linear saddle-boundary center, we show that it can have at most one limit cycle.