On determining the homological Conley index of Poincar\'e maps in autonomous systems

TL;DR

This paper presents a theorem for computing the homological Conley index of isolated invariant sets of Poincaré maps in autonomous systems, enabling analysis without detailed map information.

Contribution

The paper introduces a new theorem linking the homological Conley index to matrix reductions, simplifying computations in autonomous dynamical systems.

Findings

The homological Conley index can be obtained from a matrix derived from homology classes.

The theorem applies without requiring explicit Poincaré map values.

Special case results connect to previous work on periodic non-autonomous systems.

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.