{
  "id": "2106.14293",
  "version": "v1",
  "published": "2021-06-27T17:31:24.000Z",
  "updated": "2021-06-27T17:31:24.000Z",
  "title": "On determining the homological Conley index of Poincaré maps in autonomous systems",
  "authors": [
    "Roman Srzednicki"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "A theorem on computation of the homological Conley index of an isolated invariant set of the Poincar\\'e map associated to a section in a rotating local dynamical system $\\phi$ is proved. Let $(N,L)$ be an index pair for a discretization $\\phi^h$ of $\\phi$, where $h>0$, and let $S$ denote the invariant part of $N\\setminus L$; it follows that the section $S_0$ of $S$ is an isolated invariant set of the Poincar\\'e map. The theorem asserts that if the sections $N_0$ of $N$ and $L_0$ of $L$ are ANRs, the homology classes $[u_j]$ of some cycles $u_j$ form a basis of $H(N_0,L_0)$, and for some scalars $a_{ij}$, the cycles $u_j$ and $\\sum a_{ij}u_i$ are homologous in the covering pair $(\\widetilde N,\\widetilde L)$ of $(N,L)$ and the homology relation is preserved in $(\\widetilde N,\\widetilde L)$ under the transformation induced by $\\phi^t$ for $t\\in [0,h]$ then the homological Conley index of $S_0$ is equal to the Leray reduction of the matrix $[a_{ij}]$. In particular, no information on the values of the Poincar\\'e map or its approximations is required. In a special case of the system generated by a $T$-periodic non-autonomous ordinary differential equation with rational $T/h>1$, the theorem was proved in the paper M.\\,Mrozek, R.\\,Srzednicki, and F.\\,Weilandt, SIAM J. Appl. Dyn. Syst. 14 (2015), 1348-1386, and it motivated a construction of an algorithm for determining the index.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-27T17:31:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B30",
      "37B35"
    ],
    "keywords": [
      "homological conley index",
      "autonomous systems",
      "poincare map",
      "isolated invariant set",
      "periodic non-autonomous ordinary differential equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}