Set-valued numerical analysis for critical transitions

Complex systems research has been on the forefront of scientific priorities of many national and international research councils for more than a decade. Many very interesting phenomena have
been identified and explored, but the development of the underpinning mathematical theory has been lagging behind. The proposed research builds on an emerging movement in applied
mathematics which aims to provide proper mathematical bifurcation theory for the existence of early-warning signals for sudden changes in dynamical behaviour. These sudden changes are commonly
referred to as critical transitions, and have been reported by applied scientists in various contexts. Practical implications for the existence of such early-warning signals are far reaching, since these would enable the development of better control strategies to avoid or diminish the effect of catastrophes.

In this project, techniques for the numerical study of critical transitions will be developed. In particular, it will provide new techniques for the approximation of invariant sets. The research will be based on very recent results that make a representation of such invariant sets as functions in a Banach space possible. One of the main advantages of this new approach is that it can be used to study bifurcations, in contrast to grid-cell discretisations, which is the current state-of-the-art for the computation of invariant objects. The specific numerical studies on random systems with bounded noise will lead to insights into how an early-warning can be given should a dynamical system approach a bifurcation point.

The urgent need and timeliness of the overarching topic of critical transitions and tipping points is exemplified by the huge popular and academic interest in recent books and publications on this topic. It is also illustrated by the uniformly very positive response to the interdisciplinary workshop on this topic, held at Imperial College London last year. This meeting brought together, for the first time, a broad range of applied mathematicians and applied scientists who will guide this development in the upcoming years. The proposed research concerns a major step towards an applicability of the relevant mathematical discipline of bifurcation theory in this context; in particular, it will contribute to the development of early-warning signals, which are computational measurements that are able to indicate points where the behaviour of real world applications is about to change drastically. The societal and economic impact of such studies cannot be underestimated, since the knowledge of such tipping points will allow human interaction in order to attenuate the expected consequences. As identified at the meeting last year, practical examples can be found in many different areas and include epileptic seizures, stock market collapses, power systems blackouts, earthquakes, and climate.

This work will also provide new techniques to approximate invariant objects of dynamical systems. Set-valued computational methods have been used in form of grid-cell discretisations in the last twenty years. The proposed research will have major implications for the practicability of numerical methods in this context, since the developed algorithms will significantly outperform the current state-of-the-art methods. In many applications, new technologies are needed that make efficient use of computational resources. This is particularly important when algorithms need make quick decisions based on observed data. Typical applications include air traffic control, autoland systems for aircrafts and road-vehicle driver assistance systems, where algorithms need to compute controlled invariant sets. In contrast to the above mentioned studies in bifurcation theory, the developed research will not be directly applicable in these contexts, but they will provide a new basis that is crucial for further high-quality research in this field.