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.

On numerical verification methods to construct local Lyapunov functions around non-hyperbolic equilibria for two-dimensional cases

“…Then, if we can choose q (3) (u) such that au T d (3) (u) has a negative value for an arbitrary u ̸ = 0, the function L is a local Lyapunov function around 0. If we cannot select such a q (3) (u), we aim to determine q (3) (u) such that u T d (3) (u) = 0 holds. We determine q (i) (u) in a similar manner.…”

“…In general, the construction of Lyapunov functions is difficult. However, there exist numerical verification methods for constructing local Lyapunov functions in quadratic form around hyperbolic equilibrium [1], as well as around nonhyperbolic equilibrium in two-dimensional cases [3]. The method proposed in [3] is based on the normal form theory of dynamical systems.…”

“…Although [3] only mentioned transformation by up to third degree polynomials, in reality we can also apply polynomials of degree four or higher. In this paper, we show that a polynomial of up to an arbitrary degree can be applied to derive a non-linear transformation using the method proposed in [3] for the two-dimensional problems specified below.…”

“…Although [3] only mentioned transformation by up to third degree polynomials, in reality we can also apply polynomials of degree four or higher. In this paper, we show that a polynomial of up to an arbitrary degree can be applied to derive a non-linear transformation using the method proposed in [3] for the two-dimensional problems specified below. This indicates that we can construct a local Lyapunov function using the methods in [3] for almost any case where the original system has a local Lyapunov function around the equilibrium.…”

“…In this paper, we show that a polynomial of up to an arbitrary degree can be applied to derive a non-linear transformation using the method proposed in [3] for the two-dimensional problems specified below. This indicates that we can construct a local Lyapunov function using the methods in [3] for almost any case where the original system has a local Lyapunov function around the equilibrium. As a numerical example, we present a problem in which we fail to construct a local Lyapunov function us-ing the transformation by polynomials of degree three, but instead succeed using the transformation by a polynomial of degree five.…”