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Math-2125 · mathematics · Dynamical systems

Low degree rigid systems

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

For real parameters $a,b,c,d,e$, consider the polynomial vector field \[ \dot x=-y+xF(x,y),\qquad \dot y=x+yF(x,y),\qquad F=a+bx+cy+dx^2+exy. \] A limit cycle means the image of a nonconstant periodic solution which is isolated among periodic orbits. Distinct time parametrizations of the same closed curve count as one cycle; multiplicity is not included in the count. For $r>0$, the substitution $(x,y)=(r\cos\theta,r\sin\theta)$ gives $\dot\theta=1$ and \[ \frac{dr}{d\theta}=ar+(b\cos\theta+c\sin\theta)r^2 +(d\cos^2\theta+e\cos\theta\sin\theta)r^3. \] Thus a cycle corresponds to an isolated positive $2\pi$-periodic solution of this scalar equation, defined without blow-up throughout the period. The question is whether every member of the family has at most two limit cycles, and whether this bound is attained. Equivalently, is the supremum of the number of isolated positive periodic solutions, over all five parameters, equal to two?

The origin is the only equilibrium, and every periodic orbit surrounds it. Consequently the count includes all sizes of cycles, including those far from the origin, rather than only cycles appearing in a local bifurcation. If the origin is a center, its continuous family of surrounding periodic orbits is not counted as a family of limit cycles. The source records examples with two cycles; the outstanding content is a uniform global upper bound.

Short English statement

Can any cubic rigid system in this five-parameter family have more than two isolated periodic orbits?

Sources

  • S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700001 (AMR-046-0001), 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 1: Low degree rigid systems. Original source indexed by AMR-046-0001. https://arxiv.org/abs/2012.02524.

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2125-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2125

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.

Low degree rigid systems

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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