{
  "abstract": "We introduce an exact Rayleigh reduction for the direct construction and detection of turning bifurcations in nonlinear parameter-dependent equations. A generalized Rayleigh functional ordinarily provides only a scalar necessary constraint on solutions. We show that, along a suitable one-parameter transformation orbit $u=\\mathcal G_sϕ$, it can instead become the exact parameter law $λ=\\mathcal R(\\mathcal G_sϕ)$ of a genuine solution branch. Turning points are then found from nondegenerate critical points of this one-dimensional Rayleigh profile. The branch tangent automatically yields a kernel direction of the state linearization, and under the usual Fredholm, kernel-simplicity, and parameter-transversality assumptions the detected point is a simple fold. For Kirchhoff equations, amplitude scaling on bounded domains and spatial dilation on $\\mathbb R^N$ yield explicit exact branches, global parameter thresholds, and countable hierarchies of turning values. For the generalized Kirchhoff law $M_b(A)=a+bA^θ$, the branch geometry is governed by the dimension--homogeneity index $θ(N-2)-2$: an interior turning bifurcation occurs exactly when $θ(N-2)>2$. This gives a universal bifurcation trichotomy that is independent of the particular Berestycki--Lions nonlinearity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29150",
  "author_total": 1,
  "authors": [
    {
      "name": "Yavdat Il'yasov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9ej0vdph0xjg4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29150",
  "fulltext_url": null,
  "id": "9ej0vdph0xjg4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29150",
  "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.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7204948533",
  "pdf_url": "https://arxiv.org/pdf/2608.29150",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9ej0vdph0xjg4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29150",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29150"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Exact Rayleigh Reduction for Direct Detection of Turning Bifurcations",
  "title_text": "Exact Rayleigh Reduction for Direct Detection of Turning Bifurcations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/9ej0vdph0xjg4",
  "venue": "arXiv",
  "versions": []
}