Phase portraits, Lyapunov functions, and projective geometry

Abstract

We discuss two problems which grew out of an introductory differential equations class but were solved only later, each after having been put into a different context. First, how do you find a rather complicated Lyapunov function with your bare hands, without using a fully developed theory (while reconstructing the steps leading up to such a theory)? Second, how can you obtain a global picture of the phase-portrait of a dynamical system (thereby invoking ideas from projective geometry)? Since classroom experiences played an important part in the making of this paper, didactical aspects will also be discussed.

Stability Analysis of Equilibrium Point and Limit Cycle of Two-Dimensional Nonlinear Dynamical Systems—A Tutorial

“…The Lyapunov and LaSalle theorems are not the only way to prove stability of an equilibrium point [26][27][28][29] and are not the best approaches to use in some cases [30][31][32][33] in order to prove stability of an equilibrium point, given the amount of calculations required and the method not being as intuitive compared to the graphical phase plane method [34][35][36][37]. The graphical phase plane method is somehow straightforward as this is a picture-based method and it has been used in many applications [38][39][40][41][42][43][44][45][46][47][48].…”