Lyapunov stability of polynomial vector fields is undecidable

Abstract

We show that there are integers $N$ and odd $D$ such that no algorithm can decide, from the rational coefficients of a homogeneous polynomial vector field F of degree $D$ in dimension $N$, whether the origin is Lyapunov stable for $\dot{Y}=F(Y)$. This proves, for some large and unoptimized dimension and degree, a conjecture of V. I. Arnold.