{
  "id": "1108.2974",
  "version": "v2",
  "published": "2011-08-15T11:43:43.000Z",
  "updated": "2011-10-18T23:47:01.000Z",
  "title": "Bifurcations in Boolean Networks",
  "authors": [
    "Chris J. Kuhlman",
    "Henning S. Mortveit",
    "David Murrugarra",
    "V. S. Anil Kumar"
  ],
  "comment": "18 pages, 4 figures, Discrete Mathematics and Theoretical Computer Science 2011",
  "journal": "Mathematics and Theoretical Computer Science, proc, AP, 29-46, 2012",
  "categories": [
    "math.DS"
  ],
  "abstract": "This paper characterizes the attractor structure of synchronous and asynchronous Boolean networks induced by bi-threshold functions. Bi-threshold functions are generalizations of classical threshold functions and have separate threshold values for the transitions 0 -> 1 (up-threshold) and 1 -> 0 (down-threshold). We show that synchronous bi-threshold systems may, just like standard threshold systems, only have fixed points and 2-cycles as attractors. Asynchronous bi-threshold systems (fixed permutation update sequence), on the other hand, undergo a bifurcation: when the difference \\Delta of the down- and up-threshold is less than 2 they only have fixed points as limit sets. However, for \\Delta >= 2 they may have long periodic orbits. The limiting case of \\Delta = 2 is identified using a potential function argument. Finally, we present a series of results on the dynamics of bi-threshold systems for families of graph classes.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-10-18T23:47:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boolean networks",
      "bifurcation",
      "bi-threshold functions",
      "fixed points",
      "standard threshold systems"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.2974K"
    }
  }
}