{
  "id": "2310.08947",
  "version": "v1",
  "published": "2023-10-13T08:31:53.000Z",
  "updated": "2023-10-13T08:31:53.000Z",
  "title": "On network dynamical systems with a nilpotent singularity",
  "authors": [
    "Hildeberto Jardón-Kojakhmetov",
    "Christian Kuehn"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Network dynamics is nowadays of extreme relevance to model and analyze complex systems. From a dynamical systems perspective, understanding the local behavior near equilibria is of utmost importance. In particular, equilibria with at least one zero eigenvalue play a crucial role in bifurcation analysis. In this paper, we want to shed some light on nilpotent equilibria of network dynamical systems. As a main result, we show that the blow-up technique, which has proven to be extremely useful in understanding degenerate singularities in low-dimensional ordinary differential equations, is also suitable in the framework of network dynamical systems. Most importantly, we show that the blow-up technique preserves the network structure. The further usefulness of the blow-up technique, especially with regard to the desingularization of a nilpotent point, is showcased through several examples including linear diffusive systems, systems with nilpotent internal dynamics, and an adaptive network of Kuramoto oscillators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-13T08:31:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34E15",
      "34A26",
      "58K45"
    ],
    "keywords": [
      "network dynamical systems",
      "nilpotent singularity",
      "low-dimensional ordinary differential equations",
      "analyze complex systems",
      "zero eigenvalue play"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}