### Conley index at infinity

#### 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 PoincarĂ© 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 a normalizability condition, vector fields on $X$ are mapped to vector fields on the PoincarĂ© hemisphere whose associated flows leave the equator invariant. The dynamics on the equator reflects the dynamics at infinity, but are 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 an invariant set at infinity of isolated invariant dynamical complement. Through the construction of an extended phase space together with an extended flow, we are able to adapt the Conley index techniques and prove the existence of connections to such non-isolated invariant sets.

#### Keywords

Dynamics at infinity; blow up; heteroclinic solutions; Conley index

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