{
  "id": "2409.18681",
  "version": "v1",
  "published": "2024-09-27T12:11:47.000Z",
  "updated": "2024-09-27T12:11:47.000Z",
  "title": "Pseudometrics for scalable data-driven comparisons of nonlinear dynamical systems",
  "authors": [
    "Bryan Glaz"
  ],
  "categories": [
    "math.DS",
    "cs.SY",
    "eess.SY",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Novel solutions for pseudometrics quantifying deviation from topological conjugacy between dynamical systems are presented. Deviation from conjugacy is quantified in a Pareto optimal sense that accounts for spectral properties of Koopman operators as well as trajectory geometry. Theoretical justification is provided for computing such pseudometrics in Koopman eigenfunction space rather than observable space. Furthermore, it is shown deriving the pseudometrics from unitary transformations is sufficient to recover a value of zero if two systems are topologically conjugate. Therefore the pseudometrics for quantifying deviation from conjugacy are based on analytical solutions for unitary transformations in Koopman eigenfunction space. Finally, geometric considerations for the Pareto optimality problem associated with deviation from conjugacy are used to develop pseudometrics that account for all possible solutions given just two Pareto points based on analytical solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-27T12:11:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlinear dynamical systems",
      "scalable data-driven comparisons",
      "pseudometrics",
      "koopman eigenfunction space",
      "unitary transformations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}