Rational Lyapunov Functions and Stable Algebraic Limit Cycles

The main goal of this technical note is to show that the class of systems described by a planar differential equation having a rational proper Lyapunov function has asymptotically stable sets which are either locally asymptotically stable equilibrium points, stable algebraic limit cycles or asymptotically stable algebraic graphics. The use of the Zubov equation is then an adapted tool to investigate the study of an upper bound on the number of stable limit cycles and asymptotically stable graphics and their relative positions for this class of systems.