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Math-2127 · mathematics · Dynamical systems

Systems with homogeneous components II

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Let $a,b,c,d,e,f\in\mathbb R$ and consider \[ \dot x=ax+by,\qquad \dot y=cx^3+dx^2y+exy^2+fy^3. \] A limit cycle is an isolated nonconstant periodic orbit of this autonomous system. Different starting points on one closed orbit represent the same cycle. The first question asks whether the supremum of the number of such cycles, over all parameter choices, equals two. This is a global count in $\mathbb R^2$, including cycles not obtainable from small perturbations of an equilibrium.

The source singles out the one-parameter subfamily \[ \dot x=y,\qquad \dot y=-x^3+dx^2y+y^3, \qquad d\in\mathbb R. \] Its second question asks for a simple proof that this system has at most one limit cycle for each $d$, if that assertion is true. Here "uniqueness" is conditional on existence; it does not assert that every real parameter yields a periodic orbit. A proof should cover the entire parameter line, including the intervals between the partial existence and nonexistence ranges in the source.

Both questions concern isolation among periodic orbits, so a center with a continuum of closed trajectories does not supply infinitely many cycles. The special subfamily offers a more concrete first test of the unequal-degree homogeneous-component problem, but resolving it alone does not prove the two-cycle bound for the full six-parameter family.

Short English statement

Determine whether the full linear–cubic family has at most two limit cycles, and whether the displayed subfamily has at most one.

Sources

  • S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700003 (AMR-046-0003), 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 3: Systems with homogeneous components II. Original source indexed by AMR-046-0003. https://arxiv.org/abs/2012.02524.

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2127-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2127

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 II

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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