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Math-2690 · mathematics · Dynamical systems

Exact maximal limit-cycle count for quintic classical Lienard systems

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Over all real polynomials F of degree at most five, is the maximum number of isolated periodic orbits of $x-prime=y-F(x)$, $y-prime=-x$ exactly two, without an oddness, damping-sign or small-amplitude restriction?

Short English statement

Over all real polynomials F of degree at most five, is the maximum number of isolated periodic orbits of $x-prime=y-F(x)$, $y-prime=-x$ exactly two, without an oddness, damping-sign or small-amplitude restriction?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2690-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2690

Manuscript 1 (family 143, 2026-09-24): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Exact maximal limit-cycle count for quintic classical Lienard systems

OpenAI's claim

A classical Liénard system with arbitrary real primitive polynomial of degree at most five is claimed to have at most two isolated periodic orbits, and two are claimed to occur. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For any real polynomial $F$ of degree at most five, consider \[ \dot x=y-F(x),\qquad\dot y=-x. \] OpenAI claims at most two limit cycles in $\mathbb R^2$, with examples attaining two. A cycle means the geometric image of a nonconstant periodic trajectory isolated among periodic orbits; multiplicity and stability do not affect the count. Periodic trajectories of a center are excluded.

Outline of the argument

Manuscript 1 (family 143, 2026-09-24): A quadratic midpoint model tracks fitted slope and curvature by transport equations. Endpoint variation, Riccati linearization, and a Schwarzian identity control the width dependence of the model coefficients, allowing the two-cycle argument without the odd-polynomial restriction of the older theorem.

Formal verification

For family 143, the scope page documents the following selections. The selection concerns degree-at-most-five Liénard systems: two isolated cycles are the maximum and that maximum occurs. It is not the arbitrary-degree planar polynomial bound. The documented statement covers the main claim of manuscript 1. Comparator files: QuinticLienard.lean. Declarations: OAI.QuinticLienard.main.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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