{
  "id": "1901.08701",
  "version": "v1",
  "published": "2019-01-25T00:23:55.000Z",
  "updated": "2019-01-25T00:23:55.000Z",
  "title": "Regularisation for Planar Vector Fields",
  "authors": [
    "Nathan Duignan",
    "Holger Dullin"
  ],
  "comment": "30 pages, 7 figures, Preprint",
  "categories": [
    "math.DS"
  ],
  "abstract": "This paper serves as a first foray on regularisation for planar vector fields. Motivated by singularities in celestial mechanics, the block regularisation of a generic class of degenerate singularities is studied. The paper is concerned with asymptotic properties of the transition map between a section before and after the singularity. Block regularisation is reviewed before topological and explicit conditions for the $ C^0 $-regularity of the map are given. Computation of the $ C^1 $-regularisation is reduced to summing residues of a rational function. It is shown that the transition map is in general only finitely differentiable and a method of computing the map is conveyed. In particular, a perturbation of a toy example derived from the 4-body problem is shown to be $ C^{4/3} $. The regularisation of all homogeneous quadratic vector fields is computed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-25T00:23:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C10",
      "37C15",
      "37C25",
      "70F10",
      "70F15"
    ],
    "keywords": [
      "planar vector fields",
      "transition map",
      "block regularisation",
      "homogeneous quadratic vector fields",
      "singularity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}