{
  "id": "2404.02352",
  "version": "v1",
  "published": "2024-04-02T22:54:16.000Z",
  "updated": "2024-04-02T22:54:16.000Z",
  "title": "A remark on omega limit sets for non-expansive dynamics",
  "authors": [
    "Alon Duvall",
    "Eduardo D. Sontag"
  ],
  "comment": "8 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper, we study systems of time-invariant ordinary differential equations whose flows are non-expansive with respect to a norm, meaning that the distance between solutions may not increase. Since non-expansiveness (and contractivity) are norm-dependent notions, the topology of $\\omega$-limit sets of solutions may depend on the norm. For example, and at least for systems defined by real-analytic vector fields, the only possible $\\omega$-limit sets of systems that are non-expansive with respect to polyhedral norms (such as $\\ell^p$ norms with $p =1$ or $p=\\infty$) are equilibria. In contrast, for non-expansive systems with respect to Euclidean ($\\ell^2$) norm, other limit sets may arise (such as multi-dimensional tori): for example linear harmonic oscillators are non-expansive (and even isometric) flows, yet have periodic orbits as $\\omega$-limit sets. This paper shows that the Euclidean linear case is what can be expected in general: for flows that are contractive with respect to any strictly convex norm (such as $\\ell^p$ for any $p\\not=1,\\infty$), and if there is at least one bounded solution, then the $\\omega$-limit set of every trajectory is also an omega limit set of a linear time-invariant system.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-02T22:54:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "omega limit set",
      "non-expansive dynamics",
      "example linear harmonic oscillators",
      "time-invariant ordinary differential equations",
      "linear time-invariant system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}