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Abstract
Although, bifurcation theory of equations with autonomous and periodic time dependence is a major object of research in the study of dynamical systems since decades, the notion of a nonautonomous bifurcation is not yet established. This article contains an approach to overcome this deficit in the context of nonautonomous difference equations. Based on special notions of attractivity and repulsivity, nonautonomous bifurcation phenomena are studied. We obtain generalizations of the well-known one-dimensional transcritical and pitchfork bifurcation.
Acknowledgements
The author wishes to thank two anonymous referees for their suggestions leading to an improvement of this paper. Research supported by the “Graduiertenkolleg: Nichtlineare Probleme in Analysis, Geometrie und Physik” (GK 283) financed by the DFG and the State of Bavaria.
Notes
†This article is dedicated to Professor Bernd Aulbach in recognition for his consequent support in Augsburg and his numerous scientific achievements. He was a very talented and humane teacher with an ability to motivate his students by showing real interest in both the person and the mathematics. As a scientist, he strongly influenced with various ideas the research on the qualitative theory of nonautonomous dynamical systems. He will stay in my memory forever.