Global asymptotic stability of homogeneous polynomial vector fields is undecidable

Abstract

We study the decision problem of global asymptotic stability (GAS) of the origin for homogeneous polynomial vector fields with rational coefficients. In the first part of the paper we give, for every fixed dimension $n$ and odd degree $2k+1$, a computable map which sends a homogeneous vector field of degree $2k+1$ on $\mathbb{R}^n$ to a homogeneous cubic vector field on $\mathbb{R}^m$, $m=\binom{n+k-1}{k}$, and preserves both GAS and Lyapunov stability. The map passes to new variables $y_α=x^α$ (one for each monomial $x^α$ of degree $k$), in which the vector field becomes cubic, and adds a cubic penalty term transverse to the set of points of the form $(x^α)_{|α|=k}$; the main point is that this controls every initial state of the new system. As a consequence, the GAS problems for all degrees, for each fixed odd degree $d\ge3$, and for degree three are pairwise many-one equivalent. In the second part we prove that there are a fixed dimension $N_*$ and a fixed odd degree $d_*$ such that the set of homogeneous vector fields on $\mathbb{R}^{N_*}$ of degree $d_*$ whose origin is GAS is many-one complete. The reduction encodes an undecidable family of polynomial equations in integer unknowns into the constant term of a product of Jacobi theta series, realizes these series as outputs of a polynomial differential system with rational coefficients, and turns the existence of an integer solution into the sign of a maximal invariant average, which homogeneous GAS detects. Combining the two parts shows that GAS of rational homogeneous vector fields on $\mathbb{R}^N$ of degree $d$ is many-one complete for every odd $d\ge3$ and every sufficiently large $N$; in particular it is undecidable for cubic vector fields in a fixed dimension. Building on recent work, some analogous results are proven for Lyapunov stability of homogeneous vector fields.