Zeitschrift für Analysis und ihre Anwendungen
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Published online: 2018-03-29
Global Continuation of Homoclinic SolutionsChristian Pötzsche and Robert Skiba (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 difference 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