Conley index at infinity. (English) Zbl 1317.37021

Hell, Juliette

Topol. Methods Nonlinear Anal. 42, No. 1, 137-167 (2013).

The paper deals with flows on \(\mathbb{R}^n\) which can be compactified by adding a sphere \(S^{n-1}\) at infinity. Formally \(\mathbb{R}^n\) is identified with the upper half-sphere \(S^n_+=\{(x,z)\in\mathbb{R}^{n}\times\mathbb{R}:|x|^2+z^2=1, z>0\}\). The flow on \(\mathbb{R}^n\) corresponds to a flow on \(S^n_+\). It is required that, possibly after reparametrization, the flow on \(S^n_+\) extends to a flow on the closure \(\mathcal{H}\) of \(S^n_+\) in \(\mathbb{R}^{n+1}\) which leaves the equator \(\mathcal{E}=\{(x,z)\in\mathcal{H}:z=0\}\), the sphere at infinity, invariant.
For this type of flows the author develops Conley index techniques in order to investigate the existence of heteroclinic orbits from isolated invariant sets of the original flow to certain types of invariant sets at infinity.

Reviewer: Thomas J. Bartsch (Gießen)

MSC:

37B30	Index theory for dynamical systems, Morse-Conley indices
34C37	Homoclinic and heteroclinic solutions to ordinary differential equations
35B44	Blow-up in context of PDEs
37C29	Homoclinic and heteroclinic orbits for dynamical systems

Keywords:

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