{
  "id": "2203.09805",
  "version": "v1",
  "published": "2022-03-18T09:11:04.000Z",
  "updated": "2022-03-18T09:11:04.000Z",
  "title": "Stability Indices of Non-Hyperbolic Equilibria",
  "authors": [
    "Alexander Lohse"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We consider families of systems of two-dimensional ordinary differential equations with the origin $0$ as a non-hyperbolic equilibrium. For any number $a \\in (-\\infty, +\\infty)$ we show that it is possible to choose a parameter in these equations such that the stability index $\\sigma(0)$ is precisely $\\sigma(0)=a$. In contrast to that, for a hyperbolic equilibrium $x$ it is known that either $\\sigma(x)=-\\infty$ or $\\sigma(x)=+\\infty$. Furthermore, we discuss a system with an equilibrium that is locally unstable but seems to be globally attracting, highlighting some subtle differences between the local and non-local stability indices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-18T09:11:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stability index",
      "non-hyperbolic equilibrium",
      "two-dimensional ordinary differential equations",
      "non-local stability indices",
      "subtle differences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}