Limit Cycles of Continuous–Discontinuous Piecewise Linear Hamiltonian Systems in ℝ ² Separated by the Curve <i>y</i> =sin <i>x</i>

ABSTRACT These last decades piecewise differential systems have been studied intensively, mainly due to their applications. Inside the study of the dynamics of these differential systems, the limit cycles, that is, the isolated periodic orbits inside the set of all periodic orbits of the system, play a main role. We consider continuous–discontinuous piecewise differential systems in the plane separated by the analytic curve and formed by two arbitrary linear Hamiltonian systems: continuous in the sense that the first component of the two Hamiltonian systems coincide on the curve and discontinuous in the sense that the second component of the two Hamiltonian systems are distinct on the curve . We prove that such piecewise differential systems can have four limit cycles.