{
  "abstract": "A global classification of the asymptotic oscillatory dynamics is established for a symmetric zero-divergence class of three-dimensional piecewise-linear Filippov systems, up to a set of initial conditions of zero Lebesgue measure. The affine fields are related by an involution, and the switching plane contains a visible--visible two-fold. In canonical coordinates, the eigenvalues are \\(μ\\pm i\\) and \\(-2μ\\), while \\(H\\) measures the focal-line inclination. For every \\(μ>0\\), a simple-period crossing cycle exists if and only if \\(H\\in\\mathcal I_μ\\), and is unique, symmetric, hyperbolic, and orbitally asymptotically stable. Its half-period parametrizes \\(\\mathcal I_μ\\) and determines the crossing points, period, and Floquet multipliers. Global dissipation excludes crossing cycles with any higher number of crossings and makes the classified cycle the \\(ω\\)-limit set of every crossing-only trajectory. If attractive sliding has no interior pseudo-equilibria, sliding is transient unless the trajectory reaches the two-fold. The initial conditions leading to the two-fold lie in a countable union of analytic surfaces and curves. Under sufficiently small perturbations in the Whitney \\(C^1\\) topology, the cycle persists as the unique simple-period crossing limit cycle and remains hyperbolic and orbitally asymptotically stable.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33034",
  "author_total": 1,
  "authors": [
    {
      "name": "Samuel Carlos S. Ferreira"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bvmceetj7ztfq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33034",
  "fulltext_url": null,
  "id": "bvmceetj7ztfq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33034",
  "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.DS"
    ],
    [
      "openalex",
      "2608|Advanced Differential Equations and Dynamical Systems"
    ]
  ],
  "openalex": "W7214744523",
  "pdf_url": "https://arxiv.org/pdf/2609.33034",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bvmceetj7ztfq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.33034",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33034"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Global classification of oscillatory dynamics in symmetric zero-divergence 3D piecewise-linear Filippov systems with a visible--visible two-fold",
  "title_text": "Global classification of oscillatory dynamics in symmetric zero-divergence 3D piecewise-linear Filippov systems with a visible--visible two-fold",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/bvmceetj7ztfq",
  "venue": "arXiv",
  "versions": []
}