{
  "id": "2506.18099",
  "version": "v1",
  "published": "2025-06-22T17:04:20.000Z",
  "updated": "2025-06-22T17:04:20.000Z",
  "title": "Canard cycles of non-linearly regularized piecewise smooth vector fields",
  "authors": [
    "Peter De Maesschalck",
    "Renato Huzak",
    "Otavio Henrique Perez"
  ],
  "comment": "27 pages, 12 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "The main purpose of this paper is to study limit cycles in non-linear regularizations of planar piecewise smooth systems with fold points (or more degenerate tangency points) and crossing regions. We deal with a slow fast Hopf point after non-linear regularization and blow-up. We give a simple criterion for upper bounds and the existence of limit cycles of canard type, expressed in terms of zeros of the slow divergence integral. Using the criterion we can construct a quadratic regularization of piecewise linear center such that for any integer $k>0$ it has at least $k+1$ limit cycles, for a suitably chosen monotonic transition function $\\varphi_k:\\mathbb{R}\\rightarrow\\mathbb{R}$. We prove a similar result for regularized invisible-invisible fold-fold singularities of type II$_2$. Canard cycles of dodging layer are also considered, and we prove that such limit cycles undergo a saddle-node bifurcation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-06-22T17:04:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34D15"
    ],
    "keywords": [
      "regularized piecewise smooth vector fields",
      "non-linearly regularized piecewise smooth vector",
      "canard cycles",
      "chosen monotonic transition function",
      "limit cycles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}