OP-00708 Solution claimed

A uniform limit-cycle bound for quadratic planar dynamics

Consider the planar systems $$\dot x = P(x,y), \qquad \dot y = Q(x,y), \qquad P, Q \in \mathbb{R}[x,y], \quad \max(\deg P, \deg Q) \le 2.$$ A limit cycle is an isolated periodic orbit, counted once as a geometric curve, without multiplicity. The second part of Hilbert's sixteenth problem concerns the maximal number and relative position of the limit cycles of planar polynomial systems of a given degree. Quadratic systems with four limit cycles were constructed by Chen and Wang and by Shi.

Problem. Is there an integer $H_2 < \infty$ such that every such system has at most $H_2$ distinct limit cycles in $\mathbb{R}^2$? The bound must be independent of all coefficients of $P$ and $Q$.

Source: Hilbert (1900; English translation 1902), the second part of Problem 16, for every degree. Stated for quadratic systems by AIM — Open Applied Problems.

Imported from github.com/MColbrook/AIM.

Classification

Primary: DS — Dynamical Systems
Secondary: CA
Keywords: AIM Open Applied Problems, Hilbert's sixteenth problem, limit cycles, quadratic vector fields

References

@article{hilbert1902,
  author  = {Hilbert, David},
  title   = {Mathematical problems},
  journal = {Bull. Amer. Math. Soc.},
  volume  = {8},
  number  = {10},
  year    = {1902},
  pages   = {437--479},
  doi     = {10.1090/S0002-9904-1902-00923-3},
}
@article{chen1979,
  author  = {Chen, Lan Sun and Wang, Ming Shu},
  title   = {The relative position, and the number, of limit cycles of a quadratic differential system},
  journal = {Acta Math. Sinica},
  volume  = {22},
  year    = {1979},
  pages   = {751--758},
  note    = {In Chinese},
}
@article{shi1980,
  author  = {Shi, Songling},
  title   = {A concrete example of the existence of four limit cycles for plane quadratic systems},
  journal = {Sci. Sinica},
  volume  = {23},
  year    = {1980},
  pages   = {153--158},
}
@article{gasull2025,
  author        = {Gasull, Armengol and Santana, Paulo},
  title         = {A note on {H}ilbert 16th problem},
  journal       = {Proc. Amer. Math. Soc.},
  volume        = {153},
  number        = {2},
  year          = {2025},
  pages         = {669--677},
  doi           = {10.1090/proc/17116},
  eprint        = {2407.13465},
  archivePrefix = {arXiv},
}
@misc{aim-quadratic-hilbert-sixteen,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{A uniform limit-cycle bound for quadratic planar dynamics}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/164-quadratic-hilbert-sixteen.md},
  note         = {Entry 164 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

Uniform bounds for planar polynomial limit cycles v1 GitHub ↗

OpenAI

For every degree, we prove that the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on that degree. This establishes the uniform boundedness assertion in the second part of Hilbert's sixteenth problem. The proof uses separation of asymptotic expansions on nested complex domains and a finite-dimensional counting argument.

Full resolution · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Solution claimed	—	Recorded here: math @ 1581d4c, build/main.tex.

Citing this problem

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OP-00708, "A uniform limit-cycle bound for quadratic planar dynamics", clopen.solutions.