Global asymptotic stability of homogeneous polynomial vector fields is undecidable

@misc{shevchenko2026global,
  title = {{Global asymptotic stability of homogeneous polynomial vector fields is undecidable}},
  author = {Ivan O. Shevchenko and Jun Liu and Xinzhi Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12434},
  eprint = {2610.12434},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12434},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Global asymptotic stability of homogeneous polynomial vector fields is undecidable
AU  - Ivan O. Shevchenko
AU  - Jun Liu
AU  - Xinzhi Liu
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12434
UR  - https://arxiv.org/abs/2610.12434
AB  - 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 ℝ^n to a homogeneous cubic vector field on ℝ^m, m=binomn+k−1k, 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≥3, 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 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 ℝ^N of degree d is many-one complete for every odd d≥3 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.
ER  -