New Challenges in Nonautonomous Bifurcation Theory

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The technological and economical development of our society has generated the need to deal with very complex systems that require an accurate level of understanding. The present crisis of the financial markets, which is partially due to the failure of instruments to cope with volatility, and weather phenomena associated to climate change such as El Nio, are examples of dynamical processes with a deep economic impact that require sophisticated models to take nonautonomous (i.e., time-dependent or random) influences in form of changes of parameters into account.The main goal of this research project is to develop the qualitative theory of nonautonomous dynamical systems from a mathematical viewpoint in directions beyond the traditional setting which are highly relevant to the applied sciences. In particular, it aims at characterising and classifying qualitative changes of such systems and developing computational tools for their analysis. The availability of a universally applicable bifurcation theory will provide computational measurements which are able to indicate points where the systems behaviour is about to change drastically, with enormous possible applications including the prediction of (catastrophic) events, for instance, in economics (stock markets), environmental studies (climate change modelling) or health care studies (seizure prediction).