{
  "abstract": "Two-stage contagion, in which adoption requires two successive contacts with adopters, combined with population turnover produces bistability and relaxation oscillations in Katriel's differential-equation model, but no chaos. We derive from the elementary reactions of the process, without reference to the differential equations, a discrete-time model in which the population is updated in rounds of length $τ$, as in elections, meetings or turns of a game: each individual makes at most one transition per round, events compete as exponential risks, and adoptions take effect from the next round. The model is a Markov chain whose large-population limit is a map $F_τ$; Katriel's equations are recovered as $τ\\to0$. Taking the probability $p$ of adoption after a single contact as bifurcation parameter, with rates and round length fixed, $F_τ$ goes from extinction through a window of chaos to a stable equilibrium; for $β/γ=46$, $μ/γ=0.002$, $γτ=5.5$ the window is $0.023\\le p\\le0.056$, bounded by the invasion threshold and by a boundary crisis beyond a subcritical Neimark--Sacker point. One-stage contagion ($p=1$), $p=0$ and permanent adoption showed no chaos. The chaotic attractor is essentially unique, with one positive Lyapunov exponent and an outbreak--recharge geometry. In the same window Katriel's equations have a periodic relaxation cycle with troughs of $10^{-44}$--$10^{-164}$: the round structure does not create the oscillation but turns it into chaos and shrinks the oscillatory range in $p$ from $(0.022,0.41)$ to $(0.022,0.056)$. The chaos is therefore not an artefact of discretising a stable state, but a property of two-stage contagion with synchronous updating.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08353",
  "author_total": 3,
  "authors": [
    {
      "name": "Francisco J. Muñoz"
    },
    {
      "name": "Juan Carlos Nuño"
    },
    {
      "name": "Luca Meacci"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5dss1eckyners.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08353",
  "fulltext_url": null,
  "id": "5dss1eckyners",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08353",
  "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.DS"
    ],
    [
      "arxiv",
      "nlin.CD"
    ],
    [
      "arxiv",
      "physics.soc-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08353",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5dss1eckyners.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:30Z",
      "native_id": "2611.08353",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08353"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "physics",
      "origin": "arxiv:nlin.CD",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "From elementary processes to chaos: a discrete-time two-stage contagion model and its continuous limit",
  "title_text": "From elementary processes to chaos: a discrete-time two-stage contagion model and its continuous limit",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:34Z",
  "url": "https://research.draconitic.com/paper/5dss1eckyners",
  "venue": "arXiv",
  "versions": []
}