A Globally Asymptotically Stable Polynomial Vector Field with Rational Coefficients and no Local Polynomial Lyapunov Function

TL;DR

This paper presents a specific two-dimensional polynomial vector field that is globally asymptotically stable yet lacks a local polynomial Lyapunov function, challenging assumptions about stability certificates.

Contribution

It provides the first explicit example of such a polynomial vector field with rational coefficients and degree seven, demonstrating limitations of polynomial Lyapunov functions.

Findings

The vector field is globally asymptotically stable.

It does not admit a local polynomial Lyapunov function.

The example is explicit and of degree seven with rational coefficients.

Abstract

We give an explicit example of a two-dimensional polynomial vector field of degree seven that has rational coefficients, is globally asymptotically stable, but does not admit an analytic Lyapunov function even locally.