A Lyapunov function proof of Poincare´s theorem

One of the most fundamental results in analyzing the stability properties of periodic orbits and limit cycles of dynamical systems is Poincare´s theorem. The proof of this result involves system analytic arguments along with the Hartman-Grobman theorem. In this paper, using the notions of stability of sets, we construct lower semicontinuous Lyapunov functions to provide a Lyapunov function proof of Poincare´s theorem.