{
  "abstract": "Hopf bifurcations are a universal route to self-sustained oscillations in driven systems. Despite the absence of any singular stationary state, we show that time-averaged observables generically exhibit singularities at the onset of oscillations. The origin of this behavior is geometric: phase averaging over the emergent periodic attractor eliminates odd powers of the oscillation amplitude, while the squared amplitude varies smoothly with the distance from the bifurcation. Consequently, the excess of a time-averaged observable admits an integer-power expansion; observables remain finite but display discontinuities in finite-order derivatives. This yields an Ehrenfest-like hierarchy of Hopf singularities, in which the first nonanalytic derivative is determined by the lowest-order coupling between the observable and the limit-cycle waveform that survives phase averaging. The time averages of generic observables therefore exhibit kink singularities, while symmetry or geometric cancellations can suppress lower-order couplings and shift nonanalyticity to higher derivatives. We demonstrate this mechanism in chemical, electronic, and climate oscillators. Our results identify supercritical Hopf bifurcations as a universal mechanism for nonanalytic observable behavior, where singular yet non-diverging features emerge without any underlying singular stationary state.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.05194",
  "author_total": 2,
  "authors": [
    {
      "name": "Benedikt Remlein"
    },
    {
      "name": "Massimiliano Esposito"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1tqh14my7mkwe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.05194",
  "fulltext_url": null,
  "id": "1tqh14my7mkwe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.05194",
  "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",
      "cond-mat.stat-mech"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2605.05194",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1tqh14my7mkwe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:52Z",
      "native_id": "2605.05194",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.05194"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Singular Behavior of Observables at Hopf Bifurcations",
  "title_text": "Singular Behavior of Observables at Hopf Bifurcations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:41Z",
  "url": "https://research.draconitic.com/paper/1tqh14my7mkwe",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1103/865m-cxxg",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}