{
  "id": "2411.01181",
  "version": "v1",
  "published": "2024-11-02T08:47:25.000Z",
  "updated": "2024-11-02T08:47:25.000Z",
  "title": "On the dynamics of non-autonomous systems in a~neighborhood of a~homoclinic trajectory",
  "authors": [
    "Alessandro Calamai",
    "Matteo Franca",
    "Michal Pospisil"
  ],
  "comment": "62 pages, 11 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "This article is devoted to the study of a $2$-dimensional piecewise smooth (but possibly) discontinuous dynamical system, subject to a non-autonomous perturbation; we assume that the unperturbed system admits a homoclinic trajectory $\\vec{\\gamma}(t)$. Our aim is to analyze the dynamics in a neighborhood of $\\vec{\\gamma}(t)$ as the perturbation is turned on, by defining a Poincar\\'e map and evaluating fly time and space displacement of trajectories performing a loop close to $\\vec{\\gamma}(t)$. Besides their intrinsic mathematical interest, these results can be thought of as a first step in the analysis of several interesting problems, such as the stability of a homoclinic trajectory of a non-autonomous ODE and a possible extension of Melnikov chaos to a discontinuous setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-02T08:47:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34C37",
      "34A36",
      "34D10",
      "37G20",
      "37C60"
    ],
    "keywords": [
      "non-autonomous systems",
      "homoclinic trajectory",
      "space displacement",
      "unperturbed system admits",
      "perturbation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 62,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}