Periodic orbits of two particular classes of planar continuous piecewise linear differential systems with three zones separated by two parallel straight lines

Abstract We study the existence and non-existence of periodic orbits in the planar continuous piecewise linear differential systems with three zones separated by two parallel straight lines. First with the additional hypothesis that the three vector fields have no equilibrium points neither real nor virtual we prove that these differential systems have no periodic orbits. After assuming that the three linear vector fields are Hamiltonian we prove that these differential systems have either no periodic orbits or a continuum of periodic orbits. These results study the periodic orbits of the continuous piecewise linear differential systems with three zones separated by two parallel straight lines that until now were only done for the discontinuous piecewise linear differential systems with three zones separated by two parallel straight lines.