A Globally Asymptotically Stable Planar Homogeneous Polynomial Vector Field With No Polynomial Lyapunov Function

Jun Liu, Maxwell Fitzsimmons

arXiv:2607.16171v1 Primary: math.DS Secondary: eess.SY, math.OC

Abstract

We disprove the conjecture that every globally asymptotically stable homogeneous polynomial vector field admits a homogeneous polynomial Lyapunov function. The counterexample is a planar homogeneous cubic polynomial vector field with integer coefficients. It admits no positive definite homogeneous polynomial with nonpositive Lie derivative and, more strongly, no real-analytic Lyapunov function even locally. Nevertheless, it has an explicit degree-two homogeneous Lyapunov function that is radially unbounded, continuously differentiable everywhere, and smooth away from the origin. We also provide a machine-checked Lean 4 formalization of the main result.

Disclosure

“ersists on an open region of the two-parameter family (27)–(28). These results give a negative answer to the homogeneous polynomial Lyapunov converse conjecture posed in [6], [8]. ACKNOWLEDGMENT The authors used OpenAI GPT-5.6-Sol Pro to discover the counterexample and produce initial versions of the mathemati- cal arguments in Sections II–IV. OpenAI Codex (GPT-5.6-Sol Ultra) assisted with drafting the manuscript and developing the Lean 4 formalization of the ma”
PDF page 6

Classification: Substantial mathematical content or result generation
Multiplier: 10
Verified: 2026-08-23T18:57:41.500331Z

Structural counts

Pages 6 pdf

Theorems 2 source

Lemmas 4 source

Propositions 0 source

Corollaries 0 source

Definitions 1 source

Displayed equations 72 source

Bibliography entries 17 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file TAC_submission.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.