Nonautonomous Bifurcation Theory
Start date: Oct 1, 2008,
End date: Sep 30, 2010
"The mathematical concept of a dynamical system is founded on the fact that motions of many application processes are subjected to certain rules. Simple movements of a pendulum, complicated physical phenomena, chemical reactions, biological interactions (e.g., in predator-prey models) or even sociological patterns can be described via dynamical systems. Both the direct applicability of dynamical systems in numerous situations of the real life and the creation of the chaos theory in the 1960s have provided a great impetus to the theory of dynamical systems in the last decades and were a main reason for its success and popularity. For modeling real world phenomena, however, it is often inevitable to assume that the underlying rules are time-dependent, and the notion of a dynamical system is extended to this situation by the concept of a nonautonomous dynamical system. Nonautonomous situations arise quite frequently in the applied sciences, for instance, in studies of pollution spreading processes in coastal environments and climate modeling. In general, dynamical systems depend on parameters which reflect conditions influencing the system. In the dynamical bifurcation theory, the qualitative behaviour of the system under variation of these parameters is discussed. Although, the bifurcation theory for autonomous dynamical systems is a major object of research in the study of dynamical systems since decades, a corresponding theory for nonautonomous systems is still in its infancy, but in the last ten years, many renowned scientists started to think about nonautonomous bifurcation problems. The research and training project at hand aims at making major progress in the development of the nonautonomous bifurcation theory. In particular, further bifurcation patterns for low-dimensional systems are to be found, and moreover, high-dimensional systems shall be examined via reduction principles."
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