## Topological Methods in Nonlinear Analysis

### Conley index at infinity

Juliette Hell

#### 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 Poincaré 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 Poincaré 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.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 42, Number 1 (2013), 137-167.

Dates
First available in Project Euclid: 21 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461247297

Mathematical Reviews number (MathSciNet)
MR3155619

Zentralblatt MATH identifier
1317.37021

#### Citation

Hell, Juliette. Conley index at infinity. Topol. Methods Nonlinear Anal. 42 (2013), no. 1, 137--167. https://projecteuclid.org/euclid.tmna/1461247297

#### References

• A.A. Andronov, E.A. Leontovich, I.I. Gordon and A.G. Maier, Qualitative Theory of Second-order Dynamic Systems, Halsted Press Book (1973) \ref\key 2
• J.C. Artés and J. Llibre, Quadratic hamiltonian vector fields , Journal of Differential Equations (1994) \ref\key 3
• N. Ben-Gal, $\!$Grow-up Solutions and Heteroclinics to Infinity for Scalar Parabolic \romPDE's, PhD thesis, Brown University, Providence, USA (2009) \ref\key 4
• Ch. Conley, Isolated invariant set and the Morse index , CBMS Reg. Conf. Ser. Math., 38 (1978) \ref\key 5
• E.A. González Velasco, Generic properties of polynomial vector fields at infinity , Trans. Amer. Math. Soc., 143 , 201–222 (1969) \ref\key 6
• J. Hell, Conley Index at Infinity, PhD thesis, Freie Universität, Berlin, Germany (2010) \ref\key 7
• M. Izydorek and K.P. Rybakowski, On the Conley index in Hilbert spaces in the absence of uniqueness , Fund. Math., 171 (2002) \ref\key 8
• E. Kappos, The Conley index and global bifurcations. \romI: Concepts and theory, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 , 937–953 (1995) \ref\key 9 ––––, The Conley index and global bifurcations. \romII: Illustrative applications, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6 , 2491–2505 (1996) \ref\key 10 ––––, Compactified dynamics and peaking , Circuits and Systems, 2000. Proceedings. ISCAS 2000 Geneva. (2000) \ref\key 11
• Ch. McCord, Poincaré–Lefschetz duality for the homology Conley index , Trans. Amer. Math. Soc. (1992) \ref\key 12
• M. Messias, Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system , J. Phys. A (2009) \ref\key 13
• M. Mrozek and R. Srzednicki, On time-duality of the Conley index , Result. Math., 24 , 161–167 (1993) \ref\key 14
• L. Perko, Differential Equations and Dynamical Systems, Springer–Verlag (1991) \ref\key 15
• K.P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer–Verlag (1987) \ref\key 16
• D. Schlomiuk and N. Vulpe, Geometry of quadratic differential systems in the neighbourhood of infinity , J. Differential Equations, 215 , 357–400 (2005) \ref\key 17
• J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer–Verlag (1983)