Converse polynomial and analytic Lyapunov questions for homogeneous cubic fields

Must a globally asymptotically stable homogeneous cubic polynomial vector field admit a homogeneous polynomial Lyapunov certificate? A stronger question asks for a local real-analytic weak Lyapunov function.

Why it matters and what counts as progress

Why it matters: The converse would justify searching for polynomial certificates merely from dynamical stability.

What counts as progress: A converse theorem or an explicit stable homogeneous cubic field excluding all such polynomial certificates, with any analytic obstruction proved separately.

How the problem developed

1976 · Open problem [1]
Vladimir Arnold Raised the broader question of deciding asymptotic stability for polynomial vector fields with rational coefficients.
2011 · Conjecture [1]
Amir Ali Ahmadi Formulated the homogeneous polynomial Lyapunov converse that every globally asymptotically stable homogeneous polynomial field admits a homogeneous polynomial Lyapunov function.
17 Jul 2026 · Counterexample [1]
Jun Liu and Maxwell Fitzsimmons Produced a globally asymptotically stable planar homogeneous cubic field with integer coefficients that has no polynomial Lyapunov function, even locally analytic ones.

AI contribution

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

Jun Liu, Maxwell Fitzsimmons

Liu and Fitzsimmons give an explicit planar homogeneous cubic polynomial vector field with integer coefficients whose origin is globally asymptotically stable and has a positive radially unbounded C¹ homogeneous Lyapunov function, but has no positive definite homogeneous polynomial with nonpositive Lie derivative and no local real-analytic weak Lyapunov function.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (Lean 4/mathlib proof of all four Theorem II.2 items with axiom audit + human)

Human understandingFull