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.