On cyclicity in discontinuous piecewise linear near-Hamiltonian differential systems with three zones having a saddle in the central one

We obtain lower bounds for the maximum number of limit cycles bifurcating from periodic annuli of discontinuous planar piecewise linear Hamiltonian differential systems with three zones separated by two parallel straight lines, assuming that the linear differential subsystem in the region between the two straight lines, called of central subsystem, has a saddle at a point equidistant from these lines. (Obviously, the other subsystems have saddles or centers.) We prove that at least six limit cycles bifurcate from the periodic annuli of these kind of piecewise Hamiltonian differential systems, by linear perturbations. Normal forms and Melnikov functions, defined in two and three zones, are the main techniques used in the proof of the results.