Construction of local Lyapunov functions around non-hyperbolic equilibria by verified numerics for two dimensional cases

Abstract

Numerical verification methods are proposed in order to construct local Lyapunov functions around non-hyperbolic equilibria of dynamical systems described by ODEs in two dimensional space. The normal form theory in dynamical systems gives basic ideas of these methods. To prove negative definiteness of polynomials of higher degree than two, a new theorem on interval arithmetic is also proposed.