{
  "abstract": "The single-degree-of-freedom swing equation with sinusoidal coupling is locally softening: its oscillation frequency decreases with oscillation level near a stable equilibrium. Nonlinear modal interaction, however, can reverse this trend in multimachine systems. This letter gives a necessary-and-sufficient condition for strict leading-order local hardening, measured with respect to canonical action, in two symmetric three-machine operating-point families with equal or opposite power injections at the equal-inertia machines. An exact two-coordinate reduction expresses the condition in terms of a squared modal-frequency ratio and a dimensionless nonlinear coupling measure. The proof retains the induced mean shift and second harmonic and shows that the companion modal family remains locally softening. The criterion recovers both an equal-inertia hardening example and an exact large-inertia threshold. Coherent aggregation recovers the two-machine softening law, whereas the infinite-bus limit retains two dynamic rotor angles and permits hardening. An exact coherent-group extension preserves the selected periodic family, frequency, and canonical action for every finite machine count greater than two. Thus, three finite-inertia machines are the minimum permitting local hardening in the stated undamped, lossless classical model.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24760",
  "author_total": 1,
  "authors": [
    {
      "name": "Kai Sun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/85hmegjx4sa52.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24760",
  "fulltext_url": null,
  "id": "85hmegjx4sa52",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24760",
  "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",
      "cs.SY"
    ],
    [
      "arxiv",
      "eess.SY"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2208|Power System Optimization and Stability"
    ]
  ],
  "openalex": "W7223015655",
  "pdf_url": "https://arxiv.org/pdf/2610.24760",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/85hmegjx4sa52.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24760",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24760"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Three Machines Suffice for Local Hardening of Electromechanical Oscillations",
  "title_text": "Three Machines Suffice for Local Hardening of Electromechanical Oscillations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/85hmegjx4sa52",
  "venue": "arXiv",
  "versions": []
}