Aspects of Nonautonomous Bifurcations

Bifurcations describe qualitative changes in the long-term behavior of dynamical systems under parameter variation. Such changes occur at so-called critical values and it is not only of imminent importance to locate the critical values, but also to understand the precise nature of a bifurcation in oder to fully comprehend time-variant phenomena. Among the zoo of all bifurcations, those of Neimark-Sacker type are not only omnipresent in applications from e.g. the life sciences and economics, but also require an interesting mathematical machinery for their analysis. Roughly speaking, Neimark-Sacker bifurcations describe transitions from a point as object capturing the long term behavior, to a disk containing more complex dynamics. In the project at hand, we leave the classical framework of dynamical systems, where the law of evolution is constant in time. We are rather interested in problems subject to an aperiodic temporal forcing, which might be endogenous (seasonal effects) or exogenous (regulation, control). For such problems many of the classical concepts (eigenvalues, equilibria) fail, but the recent theory of nonautonomous dynamical systems provides an appropriate mathematical framework. More detailed, our goals are to understand nonautonomous versions of the Neimark-Sacker bifurcation, to develop tools required for their analysis, to identify possible new phenomena and to illustrate them by means of applied problems.

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