{
  "abstract": "We study positive limit cycles of the Abel equation $$\\dot{x}=A(t)x^{3}+B(t)x^{2},$$ where $A$ and $B$ are $2π$-periodic functions. By developing a half-period convexity method, we establish a new criterion ensuring that the equation has at most one positive limit cycle. Moreover, if such a limit cycle exists, it is hyperbolic. As applications, we consider two classes of planar polynomial systems that can be reduced to Abel equations. For a subfamily of cubic rigid systems, we prove that there is at most one limit cycle, thereby giving a partial answer to the first problem in Gasull's list of 33 open problems. For quadratic systems, we derive a new uniqueness criterion for limit cycles surrounding the origin and provide explicit parameter regions under which the criterion is satisfied.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19340",
  "author_total": 3,
  "authors": [
    {
      "name": "José Luis Bravo Trinidad"
    },
    {
      "name": "Renhao Tian"
    },
    {
      "name": "Yulin Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fn8z6vqr6yzct.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19340",
  "fulltext_url": null,
  "id": "fn8z6vqr6yzct",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19340",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2608|Advanced Differential Equations and Dynamical Systems"
    ]
  ],
  "openalex": "W7221962067",
  "pdf_url": "https://arxiv.org/pdf/2610.19340",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fn8z6vqr6yzct.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19340",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19340"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Uniqueness of Limit Cycles for Abel Equations and Planar Systems",
  "title_text": "Uniqueness of Limit Cycles for Abel Equations and Planar Systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/fn8z6vqr6yzct",
  "venue": "arXiv",
  "versions": []
}