{
  "id": "2303.14815",
  "version": "v1",
  "published": "2023-03-26T20:48:52.000Z",
  "updated": "2023-03-26T20:48:52.000Z",
  "title": "Mapping dynamical systems with distributed time delays to sets of ordinary differential equations",
  "authors": [
    "Daniel Henrik Nevermann",
    "Claudius Gros"
  ],
  "comment": "16 pages, 6 figures",
  "categories": [
    "math.DS",
    "nlin.CD"
  ],
  "abstract": "Real-world dynamical systems with retardation effects are described in general not by a single, precisely defined time delay, but by a range of delay times. It is shown that an exact mapping onto a set of $N+1$ ordinary differential equations exists when the respective delay distribution is given in terms of a gamma distribution with discrete exponents. The number of auxiliary variables one needs to introduce, $N$, is inversely proportional to the variance of the delay distribution. The case of a single delay is therefore recovered when $N\\to\\infty$. Using this approach, denoted the kernel series framework, we examine systematically how the bifurcation phase diagram of the Mackey-Glass system changes under the influence of distributed delays. We find that local properties, f.i. the locus of a Hopf bifurcation, are robust against the introduction of broadened memory kernels. Period-doubling transitions and the onset of chaos, which involve non-local properties of the flow, are found in contrast to be more sensible to distributed delays. Our results indicate that modeling approaches of real-world processes should take the effects of distributed delay times into account.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-26T20:48:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ordinary differential equations",
      "distributed time delays",
      "mapping dynamical systems",
      "distributed delay",
      "delay distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}