{
  "id": "2301.04048",
  "version": "v1",
  "published": "2023-01-10T15:58:23.000Z",
  "updated": "2023-01-10T15:58:23.000Z",
  "title": "A Sufficient Condition for the Super-linearization of Polynomial Systems",
  "authors": [
    "Mohamed-Ali Belabbas",
    "Xudong Chen"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "We provide in this paper a sufficient condition for a polynomial dynamical system $\\dot x(t) = f(x(t))$ to be super-linearizable, i.e., to be such that all its trajectories are linear projections of the trajectories of a linear dynamical system. The condition is expressed in terms of the hereby introduced weighted dependency graph $G$, whose nodes $v_i$ correspond to variables $x_i$ and edges $v_iv_j$ have weights $\\frac{\\partial f_j}{\\partial x_i}$. We show that if the product of the edge weights along any cycle in $G$ is a constant, then the system is super-linearizable. The proof is constructive, and we provide an algorithm to obtain super-linearizations and illustrate it on an example.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-10T15:58:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sufficient condition",
      "polynomial systems",
      "super-linearization",
      "weighted dependency graph",
      "edge weights"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}