Eigenpairs for the analysis of complete Lyapunov functions

A complete Lyapunov function describes the qualitative behaviour of a dynamical system: the areas where the orbital derivative vanishes and where it is strictly negative, characterise the chain recurrent set and the gradient-like flow, respectively. Moreover, its local maxima and minima show the stability properties of the connected components of the chain recurrent set. In this article, we use collocation with radial basis functions to numerically compute approximations to complete Lyapunov functions, and then localise and analyse the stability properties of the connected components of the chain recurrent set using its gradient and Hessian. In particular, we improve the estimation of the chain recurrent set and we determine the dimension and the stability properties of its connected components.