{
  "id": "2101.04214",
  "version": "v1",
  "published": "2021-01-11T22:13:41.000Z",
  "updated": "2021-01-11T22:13:41.000Z",
  "title": "On the stability of boundary equilibria in Filippov systems",
  "authors": [
    "David J. W. Simpson"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "The leading-order approximation to a Filippov system $f$ about a generic boundary equilibrium $x^*$ is a system $F$ that is affine one side of the boundary and constant on the other side. We prove $x^*$ is exponentially stable for $f$ if and only if it is exponentially stable for $F$ when the constant component of $F$ is not tangent to the boundary. We then show exponential stability and asymptotic stability are in fact equivalent for $F$. We also show exponential stability is preserved under small perturbations to the pieces of $F$. Such results are well known for homogeneous systems. To prove the results here additional techniques are required because the two components of $F$ have different degrees of homogeneity. The primary function of the results is to reduce the problem of the stability of $x^*$ from the general Filippov system $f$ to the simpler system $F$. Yet in general this problem remains difficult. We provide a four-dimensional example of $F$ for which orbits appear to converge to $x^*$ in a chaotic fashion. By utilising the presence of both homogeneity and sliding motion the dynamics of $F$ can in this case be reduced to the combination of a one-dimensional return map and a scalar function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-11T22:13:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34A36",
      "37G10"
    ],
    "keywords": [
      "exponential stability",
      "one-dimensional return map",
      "generic boundary equilibrium",
      "general filippov system",
      "leading-order approximation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}