{
  "id": "math/9910176",
  "version": "v1",
  "published": "1999-11-01T01:08:58.000Z",
  "updated": "1999-11-01T01:08:58.000Z",
  "title": "The geometry of bifurcation surfaces in parameter space. I. A walk through the pitchfork",
  "authors": [
    "Rowena Ball"
  ],
  "comment": "22 pages, 13 postscript figures. Better quality figures are available from the author. Submitted to Physica D",
  "categories": [
    "math.DS",
    "chao-dyn",
    "math-ph",
    "math.MP",
    "math.NA",
    "nlin.CD"
  ],
  "abstract": "The classical pitchfork of singularity theory is a twice-degenerate bifurcation that typically occurs in dynamical system models exhibiting Z_2 symmetry. Non-classical pitchfork singularities also occur in many non-symmetric systems, where the total bifurcation environment is usually more complex. In this paper three-dimensional manifolds of critical points, or limit-point shells, are introduced by examining several bifurcation problems that contain a pitchfork as an organizing centre. Comparison of these surfaces shows that notionally equivalent problems can have significant positional differences in their bifurcation behaviour. As a consequence, the parameter range of jump, hysteresis, or phase transition phenomena in dynamical models (and the physical systems they purport to represent) is determined by other singularities that shape the limit-point shell.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1999-11-01T01:08:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58Fxx"
    ],
    "keywords": [
      "parameter space",
      "bifurcation surfaces",
      "limit-point shell",
      "phase transition phenomena",
      "significant positional differences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1999math.....10176B"
    }
  }
}