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PreprintCC BY 4.0arXiv

Lyapunov stability of polynomial vector fields is undecidable

  • Milan Korda
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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 in dimension N of degree D, whether the origin is Lyapunov stable for Ẏ=F(Y). This proves a conjecture of V. I. Arnold. We provide a Lean formalization of the proof. The smallest dimension N for which we were able to prove undecidability is N = 5. An analogous undecidability result holds for global asymptotic stability and global exponential stability for non-homogenous vector fields. For dimension N=2 and homogenous vector fields, we establish decidability for all commonly used stability notions, building heavily on existing results.

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Cite

BibTeXRIS
BibTeX
@misc{korda2026lyapunov,
  title = {{Lyapunov stability of polynomial vector fields is undecidable}},
  author = {Milan Korda},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22058},
  eprint = {2609.22058},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22058},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Lyapunov stability of polynomial vector fields is undecidable
AU  - Milan Korda
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.22058
UR  - https://arxiv.org/abs/2609.22058
AB  - 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 in dimension N of degree D, whether the origin is Lyapunov stable for Ẏ=F(Y). This proves a conjecture of V. I. Arnold. We provide a Lean formalization of the proof. The smallest dimension N for which we were able to prove undecidability is N = 5. An analogous undecidability result holds for global asymptotic stability and global exponential stability for non-homogenous vector fields. For dimension N=2 and homogenous vector fields, we establish decidability for all commonly used stability notions, building heavily on existing results.
ER  -