{
  "abstract": "This paper concerns the planar Liénard system \\(\\dot x=y-F(x),\\ \\dot y=-x\\), where \\(F(x)\\) is a continuous piecewise linear function with exactly \\(n\\) fold points. Tonnelier [SIAM J. Appl. Math. 63 (2002)] conjectured that the maximum number of limit cycles of the system is \\(n\\) when \\(F\\) has exactly \\(n\\) fold points. The conjecture was confirmed for \\(n=1\\) and \\(n=2\\) in [J. Nonlinear Sci. 25 (2015)] and [J. Lond. Math. Soc. 113 (2026)], respectively, whereas the case \\(n\\ge3\\) remained open. In this paper, we show that the maximal number of limit cycles is at least \\(|3n-4|\\) for \\(n\\in\\mathbb N^+\\), thereby disproving Tonnelier's conjecture for \\(n\\ge3\\). The proof reveals a unified multiscale perturbation mechanism underlying the creation and coexistence of multiple limit cycles. Moreover, we prove that the number of limit cycles admits the finite upper bound \\(2^{28(n+1)^2}\\).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10050",
  "author_total": 4,
  "authors": [
    {
      "name": "Shiyu Cai"
    },
    {
      "name": "Hebai Chen"
    },
    {
      "name": "Jie Jin"
    },
    {
      "name": "Yuhuan Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f86rg5ap5tf5h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10050",
  "fulltext_url": null,
  "id": "f86rg5ap5tf5h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10050",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2608|Advanced Differential Equations and Dynamical Systems"
    ]
  ],
  "openalex": "W7221044965",
  "pdf_url": "https://arxiv.org/pdf/2610.10050",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f86rg5ap5tf5h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.10050",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10050"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Bounds on the maximum number of limit cycles of piecewise linear Lienard systems I. The continuous case",
  "title_text": "Bounds on the maximum number of limit cycles of piecewise linear Lienard systems I. The continuous case",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/f86rg5ap5tf5h",
  "venue": "arXiv",
  "versions": []
}