Limit cycles in planar discontinuous piecewise linear Hamiltonian systems with three equal sectors

In these last decades, the interest on the discontinuous piecewise differential systems has increased strongly, mainly due to their big number of applications. In their study, the existence or not of limit cycles play a main role. In this paper, we study a class of planar discontinuous piecewise differential systems composed of three equal sectors of angle 2π/3, where each sector is governed by a distinct linear Hamiltonian system. The discontinuity set consists of three rays, which are the boundaries of these three sectors. We prove that such differential systems can exhibit at most three crossing limit cycles. Furthermore, we construct an explicit example illustrating the existence of a discontinuous piecewise differential system that attains this upper bound. So, we have solved the extended 16th Hilbert problem for these classes of discontinuous piecewise differential systems.