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Math-2126 · mathematics · Dynamical systems

Systems with homogeneous components I

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Fix distinct positive odd integers $n,m$. A real polynomial $P_n$ is homogeneous of degree $n$ when $P_n(tx,ty)=t^nP_n(x,y)$ for every real $t$; require $P_n$ and $Q_m$ to be nonzero and of the indicated degrees. Consider \[ \dot x=P_n(x,y),\qquad \dot y=Q_m(x,y). \] A periodic orbit is the closed image of a nonconstant solution. It is a limit cycle if a neighborhood of that orbit contains no other periodic orbit. Let $L(n,m)$ be the supremum, over the coefficients of both homogeneous polynomials, of the number of distinct limit cycles in the whole plane. The question is whether \[ L(n,m)=\frac{n+m}{2}\qquad(n,m\text{ odd},\ n\ne m). \] In particular, the required upper bound must be independent of the sizes and signs of the coefficients and of the locations of the cycles. Cycles are counted geometrically, without their algebraic or bifurcation multiplicities.

The restriction to unequal odd degrees is part of the problem. Equal-degree homogeneity rescales periodic orbits into continuous families, and those families do not consist of isolated cycles. The source establishes examples with at least $(n+m)/2$ cycles when the degrees are unequal and odd. Hence the remaining assertion is that this construction already attains the global maximum. A proof for a neighborhood of the origin or a particular perturbation family would not establish the stated bound for all coefficients.

Short English statement

Is the known lower bound of half the sum of the two odd degrees also the exact global maximum number of limit cycles?

Sources

  • S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700002 (AMR-046-0002), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
  • S2 Gasull - Some open problems in low dimensional dynamical systems (2020), Problem environment 2: Systems with homogeneous components I. Original source indexed by AMR-046-0002. https://arxiv.org/abs/2012.02524.

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2126-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2126

Manuscript 1 (family 143, 2026-09-24): The degree-uniform finite bound restricts to the specified polynomial families and is independent of their coefficients and cycle locations. It supplies a weaker finite upper bound, without the requested sharp two-cycle or (n+m)/2 count or the special simple uniqueness proof.

Systems with homogeneous components I

OpenAI's claim

For each polynomial degree, a finite bound on the number of isolated periodic orbits of all real planar polynomial vector fields of that degree is claimed. 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 each integer $d\ge1$, OpenAI asserts existence of a finite nonnegative integer $B(d)$ satisfying \[ \#\{\text{isolated periodic-orbit images of }\dot x=P(x,y),\ \dot y=Q(x,y)\}\le B(d) \] for every pair of real polynomials $P,Q$ of total degree at most $d$. The count covers the whole plane and ignores multiplicity. The theorem is a qualitative uniform bound; it does not specify a sharp numerical value of $B(d)$.

Outline of the argument

Manuscript 1 (family 143, 2026-09-24): Nested complex domains separate the required asymptotic scales, while high real jets retain a fixed coherence order through rapidly rotating dependent scales. Proper barriers and generic tilts reduce parameters in a derivative-stable analytic class; finite-dimensional counting then turns absolute finiteness into a uniform degree bound.

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. Manuscript 1 is not among the papers selected by that page, so the family link does not formalize its main conclusion. It does not establish formal coverage of the complete problem clause.

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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