Lyapunov function proof of Poincaré's theorem

One of the most fundamental results in analysing the stability properties of periodic orbits and limit cycles of dynamical systems is Poincaré’s theorem. The proof of this result involves system analytic arguments along with the Hartman–Grobman theorem. Using the notions of stability of sets, lower semicontinuous Lyapunov functions are constructed to provide a Lyapunov function proof of Poincaré’s theorem.