New Challenges in Nonautonomous Bifurcation Theory

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).

This proposal marks new developments in bifurcation theory. The classical bifurcation theory, although successful in many cases, is irrelevant for many real world applications. Due to numerous applications of nonlinear and random dynamics and the urgent need of new concepts in these areas, the proposed research is expected to enhance significantly the scientific expertise of the United Kingdom, with the additional benefit of technological relevance to the community. The next paragraph on early-warning signals will set out the case why the proposed research on both non-adiabatic systems and random dynamical systems is beneficial for the industry, the public sector and the wider public. The availability of a bifurcation theory for nonautonomous systems on a finite time interval will provide computational measurements which are able to indicate points where the behaviour of real world applications is about to change drastically. The study of early-warning signals is currently an active area of research in the applied sciences and has direct consequences for nearly everybody. The dynamical systems approach considered in this project goes far beyond the mere analysis of time series, but will provide tools for the analysis of time series (which are observed in real world processes) in order to detect catastrophic events. The knowledge of such tipping points in the dynamical behaviour of real processes will allow human interaction in order to attenuate the expected consequences. To be more specific, the project will have impact on many real world applications which include climate modeling (improved weather forecast), medical applications (epileptic seizure or asthma attack prediction) or economics (stock markets). It will be part of the project to develop and provide the computational aspects of the theoretical results, which are needed for practical applications. After the fundamental mathematical questions have been clarified, I intend to actively communicate with people working in the applied sciences in order to apply the theoretical research results of my group, which will help to promote the direct applicability of this research. The conditions at Imperial College are outstanding for this purpose, having world-leading experts directly available in various disciplines such as climate dynamics (Grantham Institute) or medical applications. We frequently have discussions in our group with applied scientists (from Imperial College) who come to us with questions related to bifurcation theory. For instance, we have recently spoken to medical scientists, who wanted to analyse bifurcations of finite-time EEG and ECG data sets of different patients who suffered from apnoea. After these initial studies in the applied sciences, I will get into contact with the industry to ensure the direct applicability of the proposed research. The huge network of Imperial College Consultants will help me to get to know the relevant people in industry. One of Imperial College's five strategic intents is to communicate widely the significance of science in general, and the purpose and ultimate benefits of research activities in particular. Apart from the publication in peer-reviewed scientific journals, the results of the proposed research will be made accessible to the wider public via different methods, for instance by talks in schools, for the general public or in industry, or by submission of press articles.