{
  "id": "2106.15956",
  "version": "v1",
  "published": "2021-06-30T10:08:56.000Z",
  "updated": "2021-06-30T10:08:56.000Z",
  "title": "A finite atlas for solution manifolds of differential systems with discrete state-dependent delays",
  "authors": [
    "Hans-Otto Walther"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $r>0, n\\in\\mathbb{N}, {\\bf k}\\in\\mathbb{N}$. Consider the delay differential equation $$ x'(t)=g(x(t-d_1(Lx_t)),\\ldots,x(t-d_{{\\bf k}}(Lx_t))) $$ for $g:(\\mathbb{R}^n)^{{\\bf k}}\\supset V\\to\\mathbb{R}^n$ continuously differentiable, $L$ a continuous linear map from $C([-r,0],\\mathbb{R}^n)$ into a finite-dimensional vectorspace $F$, each $d_k:F\\supset W\\to[0,r]$, $k=1,\\ldots,{\\bf k}$, continuously differentiable, and $x_t(s)=x(t+s)$. The solutions define a semiflow of continuously differentiable solution operators on the submanifold $X_f\\subset C^1([-r,0],\\mathbb{R}^n)$ which is given by the compatibility condition $\\phi'(0)=f(\\phi)$ with $$ f(\\phi)=g(\\phi(-d_1(L\\phi)),\\ldots,\\phi(-d_{{\\bf k}}(L\\phi))). $$ We prove that $X_f$ has a finite atlas of at most $2^{{\\bf k}}$ manifold charts, whose domains are almost graphs over $X_0$. The size of the atlas depends solely on the zerosets of the delay functions $d_k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-30T10:08:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34K05",
      "34K43"
    ],
    "keywords": [
      "discrete state-dependent delays",
      "finite atlas",
      "differential systems",
      "solution manifolds",
      "delay differential equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}