{
  "id": "2310.14813",
  "version": "v1",
  "published": "2023-10-23T11:26:57.000Z",
  "updated": "2023-10-23T11:26:57.000Z",
  "title": "Catastrophe conditions for vector fields in $\\mathbb R^n$",
  "authors": [
    "Mike R. Jeffrey"
  ],
  "journal": "J. Phys. A: Math. Theor. 55 464006 (2022)",
  "doi": "10.1088/1751-8121/aca36c",
  "categories": [
    "math-ph",
    "math.DS",
    "math.MP"
  ],
  "abstract": "Practical conditions are given here for finding and classifying high codimension intersection points of $n$ hypersurfaces in $n$ dimensions. By interpreting those hypersurfaces as the nullclines of a vector field in $\\mathbb R^n$, we broaden the concept of Thom's catastrophes to find bifurcation points of (non-gradient) vector fields of any dimension. We introduce a family of determinants ${B}_j$, such that a codimension $r$ bifurcation point is found by solving the system ${B}_1=...={B}_r=0$, subject to certain non-degeneracy conditions. The determinants ${B}_j$ generalize the derivatives $\\frac{\\partial^j\\;}{\\partial x^j}F(x)$ that vanish at a catastrophe of a scalar function $F(x)$. We do not extend catastrophe theory or singularity theory themselves, but provide a means to apply them more readily to the multi-dimensional dynamical models that appear, for example, in the study of various engineered or living systems. For illustration we apply our conditions to locate butterfly and star catastrophes in a second order PDE.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-23T11:26:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34Cxx",
      "37Nxx",
      "35Axx",
      "53Zxx"
    ],
    "keywords": [
      "vector field",
      "catastrophe conditions",
      "classifying high codimension intersection points",
      "bifurcation point",
      "extend catastrophe theory"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "IOP"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}