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

Jun Liu; Maxwell Fitzsimmons

REVIEW 1 major objections 3 minor 17 references

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

T0 review · 1 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper gives a planar homogeneous cubic vector field that is globally asymptotically stable but has no polynomial — and indeed no local real-analytic — Lyapunov function, disproving a converse conjecture.

desk verdict A strong paper that likely kills the homogeneous polynomial Lyapunov converse conjecture, but the printed proof of Lemma III.2 has a real reciprocal-vs-conjugate gap that must be fixed. read the letter →

arxiv 2607.16171 v1 pith:VG7WSPIN submitted 2026-07-17 math.DS cs.SYeess.SYmath.OC

Jun Liu , Maxwell Fitzsimmons This is my paper · ORCID

classification math.DScs.SYeess.SYmath.OC MSC 34D2037C7593D30

keywords LyapunovfunctionshomogeneouspolynomialvectorfieldsglobalasymptoticstabilityconverseconjecturepositivetrigonometricpolynomialsFejér–RieszfactorizationFouriercoefficientboundsplanarcubicsystems

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Lyapunov functions certify stability by decreasing along trajectories, and for homogeneous polynomial vector fields it was conjectured that a polynomial certificate always exists. This paper disproves that conjecture with a planar homogeneous cubic vector field with integer coefficients whose origin is globally asymptotically stable. The field has no positive definite homogeneous polynomial with nonpositive derivative along trajectories, and a leading-term argument rules out even local real-analytic Lyapunov functions. Stability is proved instead by an explicit degree-two homogeneous function H = (x²+y²) exp(−10xy/(x²+y²)), which is positive definite, radially unbounded, C¹, and strictly decreasing along every trajectory. A machine-checked formalization of the main theorem accompanies the proof.

What carries the argument

The key machinery is the reduction to polar coordinates, which turns the field into ṙ = r³(−1 + 5 cos 2θ), θ̇ = r², and the resulting Fourier argument on the unit circle. A homogeneous polynomial candidate P of degree 2N restricts to a positive trigonometric polynomial p; writing g = p′/(2Np), the weak Lyapunov inequality forces w = 1 − 5 cos 2θ − g ≥ 0, whose Fourier coefficients satisfy |ŵ₂| ≤ ŵ₀ = 1. The strict Fejér–Riesz factorization of p gives the degree-independent bound |ĝ₂| < 1, and since ŵ₂ = −5/2 − ĝ₂, the triangle inequality yields the contradiction 5/2 < 2. The factor 5 in the angular dynamics is the forcing term that no polynomial's logarithmic derivative can compensate.

What would settle it

A direct refutation would be to find any positive definite homogeneous polynomial P, of any even degree, with L_f P ≤ 0 for the field (2)–(3); a semidefinite-programming search over coefficient vectors at increasing degree would locate one if Theorem II.2(3) were false. For the real-analytic claim, a local real-analytic V with V(0)=0, V>0 near 0, and L_f V ≤ 0 near 0 would refute item (4). At the technical level, computing a positive definite homogeneous polynomial whose normalised logarithmic derivative has |ĝ₂| ≥ 1 would falsify Lemma III.3, the load-bearing bound.

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Extended reading notes

Core claim

The central discovery is a single planar system, the homogeneous cubic vector field f₁ = 4x³ − x²y − 6xy² − y³, f₂ = x³ + 4x²y + xy² − 6y³. In polar coordinates it becomes ṙ = r³(−1 + 5 cos 2θ), θ̇ = r², so the angle increases monotonically while the radius is governed by a second-harmonic forcing. The paper proves that the origin is globally asymptotically stable, that H = (x²+y²) exp(−10xy/(x²+y²)) is a valid non-polynomial Lyapunov function with L_fH = −2(x²+y²)H < 0, and that no positive definite homogeneous polynomial P — of any degree — satisfies the weak Lyapunov inequality L_fP ≤ 0. Since the vector field is minimal in dimension and degree for this phenomenon, the result closes the h

Load-bearing premise

The argument's load-bearing premise is Lemma III.3: for every positive definite homogeneous polynomial of degree 2N, the normalised logarithmic derivative g(θ) = p′(θ)/(2N p(θ)) has second Fourier coefficient strictly below 1 in modulus; if that strict bound fails, a polynomial Lyapunov function could pass the Fourier test.

Editorial extensions

If this is right

The homogeneous polynomial Lyapunov converse conjecture is false: polynomial certificates are not guaranteed for homogeneous polynomial systems.
For this system, the explicit C¹ function H and the rational function R both prove global asymptotic stability, so non-polynomial certificates can be simpler than polynomial ones.
Any algorithm that searches only over polynomial Lyapunov functions of bounded degree cannot certify all stable homogeneous cubic systems.
The counterexample persists on an open set of the two-parameter family (27)–(28), so the failure of polynomial certificates is robust under parameter perturbation.
The real-analytic obstruction means the phenomenon is not an artifact of restricting to polynomials; no analytic certificate exists locally either.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The second-harmonic obstruction suggests a general threshold: for the family ṙ = r³(−a + b cos 2θ), θ̇ = r², polynomial Lyapunov functions may exist only when |b| is below roughly 2(a+1); establishing existence in that regime would show that absence is governed by a sharp parameter inequality.
Because the proof only uses the second Fourier coefficient of the angular forcing, analogous non-existence arguments should hold for higher-degree homogeneous fields whose polar form has a dominant 2θ component; constructing such higher-degree examples would test the generality of the mechanism.
The regularity–algebraicity coupling observed here — a C^m homogeneous function of degree m is a polynomial — suggests that smooth non-polynomial certificates occupy a distinct complexity class from algebraic ones; that distinction may matter for designing certificate-search algorithms.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The headline is that this paper gives a genuine counterexample to the homogeneous polynomial Lyapunov converse conjecture: a planar cubic homogeneous field with integer coefficients, globally asymptotically stable, with a C^1 exponential homogeneous Lyapunov function, yet no homogeneous polynomial—or even local real-analytic—Lyapunov function. If it holds up, it settles a question open since Ahmadi's 2011 thesis and the 2012 ACC paper. The polar reduction is clean, the H function is explicit and verifiable, and the Fourier-coefficient contradiction in Section III-C is genuinely clever. I checked the key expansions (6)-(7), the identity Hdot/H = -2r^2, and the leading-term argument in III-D; those all work. The rational Lyapunov function in IV-C and the parametric robustness in IV-D are nice extras. The Lean 4 formalization is claimed, but no commit hash or extraction details are in the manuscript, so it is not independently checkable from the text.

The main soft spot is real. The stress-test note is correct: in Lemma III.2 the proof defines B#(z)=z^L B(1/z) and claims roots pair as beta and 1/beta, but for real-valued q with complex Fourier coefficients the correct pairing is beta and 1/conjugate(beta). The printed factorization A = kappa B B# is therefore false for generic q. The lemma itself is a standard Fejer-Riesz result, and the proof is repairable by replacing B# with z^L conjugate(B(1/conjugate(z))) (or by citing a textbook), and the strict bound |g_2|<1 survives. But as written, the self-contained proof has a gap. A referee must insist on this correction. Everything downstream of the lemma—the contradiction in III-C and the analytic obstruction in III-D—is built on the corrected version, so the central theorem is not threatened, but the manuscript is not fully sound in its current form.

The other caveats are minor: the literature claim that the conjecture was open rests on the authors' reading of [6],[8],[9], which the reader checked; and the Lean formalization needs a stable URL or commit. The paper is worth serious review and, once the lemma proof is repaired, it should be accepted. I would bring it to a reading group and cite it once the corrected version appears.

Referee Report

1 major / 3 minor

Summary. The paper disproves the homogeneous polynomial Lyapunov converse conjecture by constructing a planar cubic homogeneous polynomial vector field (2)–(3) with integer coefficients. The origin is globally asymptotically stable, and the explicit 2-homogeneous function H in (4) satisfies L_fH = −2(x²+y²)H < 0, with H ∈ C¹(R²) ∩ C∞(R²\{0}). In polar form the system is ṁ = r³(−1+5cos2θ), θ̇ = r². The authors show that no positive definite homogeneous polynomial P satisfies L_fP ≤ 0: via Fejér–Riesz factorization and a logarithmic Fourier-coefficient bound |ĝ₂| < 1, the weak Lyapunov inequality forces |ŵ₂| ≤ 1 while ŵ₂ + ĝ₂ = −5/2, giving 5/2 ≤ |ŵ₂| + |ĝ₂| < 2. A leading-term argument extends the obstruction to local real-analytic Lyapunov functions. A two-parameter family (27)–(28) provides an open set of counterexamples, and a rational Lyapunov function is exhibited. A Lean 4 formalization is also claimed.

Significance. If the main result is correct, it settles a long-standing open question in a surprising direction: the homogeneous-polynomial converse Lyapunov theorem fails even for planar homogeneous cubics. The example is remarkably simple (integer coefficients), and the method is elegant and degree-independent. The paper also supplies a machine-checked Lean 4 formalization, which is a substantial contribution to trustworthiness, as is the explicit rational Lyapunov certificate. These features make the result significant to the systems and control community and to algebraic/geometric stability theory.

major comments (1)

[III-B, Lemma III.2 and Eq. (13)] The displayed identity (13), A(z)=z^{2L}A(1/z), is false for general real-valued trigonometric polynomials because c_{−j}=\bar c_j does not imply c_j=c_{−j}. The correct symmetry is A(z)=z^{2L}\overline{A(1/\bar z)}, so roots pair as ζ ↔ 1/\barζ, not ζ ↔ 1/ζ. Consequently the defined B^#(z)=z^L B(1/z) has reciprocal (not reciprocal-conjugate) roots, and the equality A=κ BB^# is false in general; e.g. q(φ)=1+0.5 sinφ. This invalidates the printed proof of the factorization. Since Lemma III.3's bound (14) rests on the factorized form (15), and Lemma III.3 is the load-bearing estimate in the contradiction of §III-C, the proof must be corrected. The standard Fejér–Riesz theorem is correct, and replacing B^# with \tilde B(z)=z^L\overline{B(1/\bar z)} gives a valid proof; but as printed the paper's self-contained argument is incomplete. This is a major, though locally repairable, gap.

minor comments (3)

[Abstract] “Degree-two homogeneous Lyapunov function” may be misread as a polynomial of degree two; H is 2-homogeneous but not polynomial. Suggest “2-homogeneous” or “non-polynomial 2-homogeneous.”
[IV-D] In the displayed necessary condition |b| ≤ 2(a+1), it may be worth noting explicitly that this condition is necessary only and is not claimed sufficient; the surrounding text already implies this, but a short clarifying sentence would help.
[III-D] The proof that the lowest Taylor term P_m is positive definite relies on F vanishing on an interval; the argument is correct but somewhat terse. A brief justification that F′≤0 and F≥0 with F(θ₀)=0 forces F≡0 would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main non-existence proof is self-contained and targeted at an external conjecture.

full rationale

The derivation chain is not circular. The counterexample field (2)-(3) is introduced with explicit integer coefficients, and the Lyapunov function H in (4) is an explicit homogeneous function. The stability argument is a direct calculation: equations (6)-(9) give the polar dynamics and L_f H = -2r^2H; no parameter is fitted to the conclusion. The non-existence of homogeneous polynomial Lyapunov functions (Theorem II.2(3)) is an impossibility argument over arbitrary candidate degree 2N: it assumes a positive definite homogeneous P, forms w = 1 - 5 cos 2θ - g in (17), and uses only the positivity of w plus the degree-independent Fourier bound |ĝ2| < 1 from Lemma III.3. The contradiction 5/2 ≤ |ŵ2|+|ĝ2| < 2 does not invoke the target result or adjust any constant. Lemma III.3 rests on a standard external theorem (strict Fejér-Riesz factorization, [15]) and an elementary series computation, not on this paper's conclusion or on self-citation. Section III-D reduces real-analytic candidates to their lowest-degree homogeneous term and reuses the already proven polynomial obstruction. The only self-citation ([10] in the introduction) is contextual and not load-bearing. A caveat: the elementary proof of Lemma III.2 as printed is flawed for complex Fourier coefficients (it pairs roots by reciprocals rather than reciprocal conjugates); this is a correctness gap in an otherwise standard lemma, not a circular dependency, so it is excluded from the circularity score.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters or fitted constants: the integer coefficients of (2)–(3) are the counterexample itself and are verified by direct expansion [(6)–(7)]; the parameters a > 0, b ∈ R in §IV-D are hypotheses of a conditional family theorem, not fitted values; the exponent 5 in the rational certificate (24) is verified by the derivative identity (26). The proof rests only on standard background: Lyapunov's direct method, the self-contained Fejér–Riesz lemma, standard Fourier/log-series facts, and the Taylor structure of real-analytic functions. No new entities (particles, forces, dimensions, conserved quantities) are postulated; the functions H, W, R are explicit and verified candidates.

assumptions (7)

standard math Lyapunov's direct method: a C¹ positive definite, radially unbounded V with V̇ < 0 away from the origin implies global asymptotic stability.
Applied in §III-A to H to conclude Theorem II.2(1); also used for the family H_{a,b} in §IV-D.
standard math Strict Fejér–Riesz factorization: a strictly positive real trigonometric polynomial of actual degree L factors as |Q(e^{iφ})|² with deg Q = L and Q having no zeros in the closed unit disk.
Proved in-text as Lemma III.2; underpins the bound |ĝ₂| < 1 in Lemma III.3, the engine of the non-existence proof in §III-C.
standard math Uniform convergence of the logarithmic series log(1−αe^{i2θ}) = −Σ α^k/k e^{2ikθ} for |α| < 1, and termwise differentiation.
Used inside Lemma III.3 to compute G's Fourier coefficients and obtain ĝ₂ = −(i/N)Σα_ℓ.
standard math A real-analytic function at the origin has a convergent Taylor expansion; its lowest-degree nonzero homogeneous part P_m is well defined (Krantz–Parks [16]).
Used in §III-D, equation (19), to transfer the homogeneous obstruction to local real-analytic Lyapunov functions.
standard math A positive definite homogeneous polynomial must have even degree.
Used in §III-C to write deg P = 2N.
standard math A homogeneous function that is C^m in a neighborhood of the origin is a homogeneous polynomial of degree m (Lemma IV.1, proved in the text).
Used in §IV-A to show H_k ∈ C^{k−1} \ C^k, sharpening Lemma III.1; not load-bearing for Theorem II.2.
domain assumption Global forward well-posedness of (2)–(3) for all initial conditions.
Needed for 'globally' in global asymptotic stability; supplied by the strict, radially unbounded Lyapunov function H (trajectories bounded, flow smooth), so standard rather than ad hoc.

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Pith. "Pith review of A Globally Asymptotically Stable Planar Homogeneous Polynomial Vector Field With No Polynomial Lyapunov Function." pith.science (2026). https://pith.science/paper/VG7WSPIN

@misc{pith2026260716171,
  author       = {Pith},
  title        = {Pith review of: A Globally Asymptotically Stable Planar Homogeneous Polynomial Vector Field With No Polynomial Lyapunov Function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VG7WSPIN}},
  note         = {Machine review of arXiv:2607.16171}
}

read the original 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.

Discussion (0). Continue with ORCID to comment.

Reference graph

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