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Zeitschrift für Analysis und ihre Anwendungen

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Volume 37, Issue 2, 2018, pp. 159–187
DOI: 10.4171/ZAA/1608

Published online: 2018-03-29

Global Continuation of Homoclinic Solutions

Christian Pötzsche[1] and Robert Skiba[2]

(1) Alpen-Adria Universität Klagenfurt, Austria
(2) Nicolaus Copernicus University, Torun, Poland

When extending bifurcation theory of dynamical systems to nonautonomous problems, it is a central observation that hyperbolic equilibria persist as bounded entire solutions under small temporally varying perturbations. In this paper, we abandon the smallness assumption and aim to investigate the global structure of the entity of all such bounded entire solutions in the situation of nonautonomous diff erence equations. Our tools are global implicit function theorems based on an ambient degree theory for Fredholm operators due to Fitzpatrick, Pejsachowicz and Rabier. For this we yet have to restrict to so-called homoclinic solutions, whose limit is 0 in both time directions.

Keywords: Topological degree, Fredholm operator, properness, nonautonomous dynamical system, exponential dichotomy

Pötzsche Christian, Skiba Robert: Global Continuation of Homoclinic Solutions. Z. Anal. Anwend. 37 (2018), 159-187. doi: 10.4171/ZAA/1608