{
  "id": "1612.02162",
  "version": "v1",
  "published": "2016-12-07T09:30:30.000Z",
  "updated": "2016-12-07T09:30:30.000Z",
  "title": "Rigorous numerics of tubular, conic, star-shaped neighborhoods of slow manifolds for fast-slow systems",
  "authors": [
    "Kaname Matsue"
  ],
  "comment": "45 pages, 7 figures. Put the validation (raw) code at [27]. The document will be updated there in near future",
  "categories": [
    "math.DS",
    "math.NA"
  ],
  "abstract": "We provide a rigorous numerical computation method to validate tubular neighborhoods of normally hyperbolic slow manifolds with the explicit radii for the fast-slow system \\begin{equation*} \\begin{cases} x' = f(x,y,\\epsilon), and y' =\\epsilon g(x,y,\\epsilon). & \\end{cases} \\end{equation*} Our main focus is the validation of the continuous family of eigenpairs $\\{\\lambda_i(y;\\epsilon), u_i(y;\\epsilon)\\}_{i=1}^n$ of $f_x(h_\\epsilon(y),y,\\epsilon)$ over the slow manifold $S_\\epsilon = \\{x = h_\\epsilon(y)\\}$ admitting the graph representation. In order to obtain such a family, we apply the interval Newton-like method with rigorous numerics. The validated family of eigenvectors generates a vector bundle over $S_\\epsilon$ determining normally hyperbolic eigendirections rigorously. The generated vector bundle enables us to construct a tubular neighborhood centered at slow manifolds with explicit radii. Combining rate conditions for providing smoothness of center-(un)stable manifolds, we can validate smooth tubular neighborhoods with diffeomorphic family of affine change of coordinates, as well as several extensions such as conic and star-shaped neighborhoods. Our procedure provides a systematic construction of smooth neighborhoods of slow manifolds in an explicit range $[0,\\epsilon_0]$ of $\\epsilon$ with rigorous numerics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-07T09:30:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34A26",
      "34A30",
      "37D20",
      "57R25",
      "65L11"
    ],
    "keywords": [
      "rigorous numerics",
      "fast-slow system",
      "star-shaped neighborhoods",
      "normally hyperbolic eigendirections",
      "validate smooth tubular neighborhoods"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}