Crossing limit cycles of planar discontinuous piecewise differential systems - Renteria Alva, Sonia Isabel (2025)
In this thesis, we investigate the existence and the maximum number of crossing limit cycles in certain classes of piecewise discontinuous differential systems in the plane. In the first class, we consider systems composed of linear centers separated by two concentric circles. We establish that the maximum number of limit cycles in these systems is two and provide examples showing that this number is attained. In the second class, we study piecewise linear Hamiltonian systems separated by a curve, which may be a non-regular boundary formed by a sector with an angle between $0$ and $\\pi$, or a singular and irreducible cubic algebraic curve. For these systems, we determine the upper bound for the number of crossing limit cycles. Finally, in the third class, we examine systems formed by an arbitrary planar linear center together with one of the three reversible isochronous cubic centers with a rational first integral, separated by a circle, and analyze the maximum number of limit cycles that they can exhibit.