{
  "abstract": "We construct a two-parameter family of polynomial vector fields in $\\mathbb{R}^3$ and prove that, on a nonempty open set of parameter values, every member of the family is Hurwitz and possesses two non-constant periodic orbits, one of saddle type and one attracting. This answers Problem 19 in Gasull's list of open problems on low-dimensional dynamical systems concerning the existence of non-constant periodic orbits in Hurwitz smooth vector fields on $\\mathbb{R}^3$. This also provides counterexamples to the expected non-existence of non-constant periodic orbits for Hurwitz polynomial vector fields; a question that arose from the first counterexamples to the Markus-Yamabe conjecture in the late 1990s and that was stated by Meisters in his featured MathSciNet review of the Bernat-Llibre counterexample.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09176",
  "author_total": 2,
  "authors": [
    {
      "name": "Álvaro Castañeda"
    },
    {
      "name": "Salomón Rebollo-Perdomo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4hrrfeb4905gv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09176",
  "fulltext_url": null,
  "id": "4hrrfeb4905gv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09176",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2608|Advanced Differential Equations and Dynamical Systems"
    ]
  ],
  "openalex": "W7220929188",
  "pdf_url": "https://arxiv.org/pdf/2610.09176",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4hrrfeb4905gv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09176",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09176"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "A family of Hurwitz polynomial vector fields in $\\mathbb{R}^3$ having non-constant periodic orbits",
  "title_text": "A family of Hurwitz polynomial vector fields in ℝ^3 having non-constant periodic orbits",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/4hrrfeb4905gv",
  "venue": "arXiv",
  "versions": []
}