{
  "abstract": "We study the discrete Huygens phase map on the ring $C_n$, the simplest oscillator network in which the coupling can be frustrated. Reciprocal coupling favours phase opposition between neighbours, while the edge phase differences must sum to zero modulo $2π$ around the cycle; for odd $n$ these two requirements are incompatible. The parity of $n$ determines the absolute minima of the potential. When $4\\mid n$, the critical set also contains circles of critical points, all of which meet at two completely degenerate configurations. After quotienting by the common phase, the map is a gradient step for an intrinsic potential on an $(n-1)$-torus, and the cycle spectrum gives an explicit interval of interaction strengths on which every orbit converges to a critical point. Our main result is that, on this whole interval and for every $n\\ge3$, almost every orbit converges to a uniform twisted minimum, including at interaction strengths for which the map is not invertible. When $4\\mid n$, the non-minimal points of the critical circles are treated by a centre-stable manifold argument. At the two degenerate intersections, a local blow-up reduces the problem to the cubic leading term of the potential and its restriction to the unit sphere. Orbits approaching along directions where this restriction is positive form a null set, because the cubic is harmonic in the intrinsic metric. No orbit approaches along its zero level, where the critical points are non-degenerate saddles on invariant rays. Hence the degenerate equilibria also have basins of zero measure.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19953",
  "author_total": 2,
  "authors": [
    {
      "name": "Jorge Buescu"
    },
    {
      "name": "Henrique M. Oliveira"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b11a5y79yd4th.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19953",
  "fulltext_url": null,
  "id": "b11a5y79yd4th",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19953",
  "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.DS"
    ],
    [
      "openalex",
      "1705|Nonlinear Dynamics and Pattern Formation"
    ]
  ],
  "openalex": "W7221861230",
  "pdf_url": "https://arxiv.org/pdf/2610.19953",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b11a5y79yd4th.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19953",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19953"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Almost-global convergence to twisted states in discrete Huygens rings",
  "title_text": "Almost-global convergence to twisted states in discrete Huygens rings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/b11a5y79yd4th",
  "venue": "arXiv",
  "versions": []
}