Conley Index at Infinity

TL;DR

This paper extends Conley index techniques to study heteroclinic connections involving invariant sets at infinity using Poincaré compactification, enabling analysis of non-isolated behaviors at infinity.

Contribution

It introduces a method to analyze invariant sets at infinity with Conley index, including non-isolated behaviors, via an extended phase space and flow.

Findings

Established a framework for invariant sets at infinity

Proved existence of connections to non-isolated invariant sets

Extended Conley index techniques to the Poincaré compactification context

Abstract

The aim of this paper is to explore the possibilities of Conley index techniques in the study of heteroclinic connections between finite and infinite invariant sets. For this, we remind the reader of the Poincar\'e compactification: this transformation allows to project a $n$-dimensional vector space $X$ on the $n$-dimensional unit hemisphere of $X\times \mathbb{R}$ and infinity on its $(n-1)$-dimensional equator called the sphere at infinity. Under normalizability condition, vector fields on $X$ transform into vector fields on the Poincar\'e hemisphere whose associated flows let the equator invariant. The dynamics on the equator reflects the dynamics at infinity, but is now finite and may be studied by Conley index techniques. Furthermore, we observe that some non-isolated behavior may occur around the equator, and introduce the concept of invariant sets at infinity of isolated invariant dynamical complement. Through the construction of an extended phase space together which an extended flow, we are able to adapt the Conley index techniques and prove the existence of connections to such non-isolated invariant sets.