Crossing limit cycles for discontinuous piecewise differential systems formed by linear Hamiltonian saddles or linear centers separated by a conic

Abstract

The extension of the 16th Hilbert problem to discontinuous piecewise linear differential systems asks for an upper bound for the maximum number of crossing limit cycles that such systems can exhibit. The study of this problem is being very active, especially for discontinuous piecewise linear differential systems defined in two zones and separated by one straight line. In the case that the differential systems in these zones are formed either by linear centers or linear Hamiltonian saddles it is known that there are no crossing limit cycles. However it is also known that the number of crossing limit cycles can change if we change the shape of the discontinuity curve. In this paper we study the maximum number of crossing limit cycles of discontinuous piecewise differential systems formed by either linear Hamiltonian saddles or linear centers and separated by a conic which intersect the conic in two points. For this class of discontinuous piecewise differential systems we solve the extended 16th Hilbert problem.