{
  "id": "2101.01927",
  "version": "v1",
  "published": "2021-01-06T08:52:47.000Z",
  "updated": "2021-01-06T08:52:47.000Z",
  "title": "Generalized Liénard systems, singularly perturbed systems, Flow Curvature Method",
  "authors": [
    "Jean-Marc Ginoux",
    "Dirk Lebiedz",
    "Jaume Llibre"
  ],
  "comment": "19 pages, 1 figure. arXiv admin note: text overlap with arXiv:1408.4894",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In his famous book entitled \\textit{Theory of Oscillations}, Nicolas Minorsky wrote: \"\\textit{each time the system absorbs energy the curvature of its trajectory decreases} and \\textit{vice versa}\". According to the \\textit{Flow Curvature Method}, the location of the points where the \\textit{curvature of trajectory curve}, integral of such planar \\textit{singularly dynamical systems}, vanishes directly provides a first order approximation in $\\varepsilon$ of its \\textit{slow invariant manifold} equation. By using this method, we prove that, in the $\\varepsilon$-vicinity of the \\textit{slow invariant manifold} of generalized Li\\'{e}nard systems, the \\textit{curvature of trajectory curve} increases while the \\textit{energy} of such systems decreases. Hence, we prove Minorsky's statement for the generalized Li\\'{e}nard systems. Then, we establish a relationship between \\textit{curvature} and \\textit{energy} for such systems. These results are then exemplified with the classical Van der Pol and generalized Li\\'{e}nard \\textit{singularly perturbed systems}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-06T08:52:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "flow curvature method",
      "generalized liénard systems",
      "singularly perturbed systems",
      "invariant manifold",
      "trajectory curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}