{
  "id": "1304.4317",
  "version": "v1",
  "published": "2013-04-16T03:12:54.000Z",
  "updated": "2013-04-16T03:12:54.000Z",
  "title": "On resolving singularities of piecewise-smooth discontinuous vector fields via small perturbations",
  "authors": [
    "David J. W. Simpson"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "A two-fold singularity is a point on a discontinuity surface of a piecewise-smooth vector field at which the vector field is tangent to the surface on both sides. Due to the double tangency, forward evolution from a two-fold is typically ambiguous. This is an especially serious issue for two-folds that are reached by the forward orbits of a non-zero measure set of initial points. The purpose of this paper is to explore the concept of perturbing the vector field so that forward evolution is well-defined, and characterising the perturbed dynamics in the limit that the size of the perturbation tends to zero. This concept is applied to a two-fold in two dimensions. Three forms of perturbation: hysteresis, time-delay, and noise, are analysed individually. In each case, the limit leads to a novel probabilistic notion of forward evolution from the two-fold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-04-16T03:12:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34E10",
      "37E99",
      "34F05"
    ],
    "keywords": [
      "piecewise-smooth discontinuous vector fields",
      "small perturbations",
      "resolving singularities",
      "singularity",
      "forward evolution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.4317S"
    }
  }
}