Set-valued numerical analysis for critical transitions

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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 havebeen identified and explored, but the development of the underpinning mathematical theory has been lagging behind. The proposed research builds on an emerging movement in appliedmathematics 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 commonlyreferred 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.