On the construction of Lyapunov functions with computer assistance

This paper aims at applications of Lyapunov functions as tools for analyzing concrete dynamical systems with computer assistance, even for non-gradient-like systems. We want to know concrete form of Lyapunov functions around invariant sets and their domains of definition for applying Lyapunov functions to various analysis of both continuous and discrete dynamical systems. Although there are several abstract results for the existence of Lyapunov functions, they cannot induce a systematic and concrete procedure of Lyapunov functions with explicit forms. In this paper, we present a numerical verification method which can validate Lyapunov functions with explicit forms and their explicit domains of definition, which can be applied to arbitrary dynamical systems with (hyperbolic) equilibria or fixed points. The proposed procedure provides us with a powerful validation tool for analyzing asymptotic behavior of dynamical systems.