A geometric view of extremes in dynamical systems

Predicting high impact extreme events, such as severe climatic and economic events is a major societal challenge. Using innovative mathematical techniques this proposal determines phenomenological mechanisms that lead to the occurrence of extremes, and develops a theory that can be used to predict when such events occur in physical modelling applications. Using dynamical systems theory, the proposed research will use geometrical features of the underlying mathematical models to determine the future extreme behaviour. This goes beyond certain traditional approaches such as monitoring output time series data alone.

The study of successive maxima (or minima) for stochastic processes is called Extreme Value Theory (EVT). This theory is extensively used in risk analysis to estimate probabilities of rare events and extremes, e.g. high river levels; hurricanes and market crashes. For physical systems modelled by deterministic dynamical systems, especially chaotic dynamical systems a corresponding theory of extremes is yet to be fully understood. These systems are highly sensitive and the time series of observations can be highly correlated. A key question that we address is when to modify the theory for independent, identically distributed random variables in the case of understanding extremes for deterministic systems. Conversely when are certain probabilistic limit laws (such as Poisson laws) a good description of the extreme phenomenon?

Ergodic theory approaches have been very successful in understanding the long-term evolution of these systems. Recent approaches have focused on time series observations which have a unique maxima at a distinguished point in phase space, and whose level set geometries coincide with balls in the ambient (usually Euclidean) metric. However extremes of other physically relevant functions (with geometries beyond nested balls), are also important in applications. This includes energy-like functions or wind speed functionals which play a role in measuring the destructiveness of storms. We therefore go beyond existing methodologies and develop a theory of extremes for physically relevant observable functions. We then apply this theory to explicit dynamical systems (both discrete and continuous) motivated by real-world mathematical models such as for the weather and climate.

This project is primarily about fundamental scientific research; it is an ambitious programme to develop and apply novel mathematical methods to highly complex dynamical systems. This proposal has built in clear mechanisms to transfer the developed theory to potential end-users such as those in the weather/climate industries and the insurance industry. On a broader level this project will deliver methods to understand physical systems which are either highly sensitive to initial conditions and/or have a strong time-dependency structure. We provide a theoretical and practical framework for understanding extremes and recurrence statistics in dynamical systems. This research will give immediate academic impact in the mathematical and physical sciences, such as in ergodic theory, probability theory and statistical physics. Short-term impact will be delivered to those working in practical aspects of dynamical systems, e.g. tipping points; and those working in statistical estimation. Medium-term impact will apply to those working with complex weather and climate models.

Impact will be delivered through scientific engagement and publications. Publication routes include journals in ergodic theory, probability, dynamical systems, and statistics. To maximize impact, research outputs will be communicated via a range media options (e.g. posters, video, local and national media), as well as the more specialist journals. Results will be presented by the PDRA, the PI, the Co-I and/or visiting fellows (VFs) at international conferences. Furthermore these research outputs will be disseminated via members of the Centre for Systems, Dynamical and Control (CSDC) and members of Exeter Climate Systems (XCS) and through various international contacts of the PI/Co-I and VFs.

Through a planned workshop (on modelling extremes) impact will be delivered by bringing together researchers in dynamical systems, researchers in statistics (e.g. those in weather/climate modelling), the industrial partners (Met Office, Willis Towers Watson), and other potential end-users such as Public Health England.

In the medium term the proposed project is well placed to provide economic benefits to UK society through creating methodologies that improve weather and climate forecasting, and in prediction of extreme weather events. We will engage with end-users and policymakers to help plan and mitigate damage to infrastructure and health due to severe weather events. This will be achieved by the PI's/Co-I's interactions with XCS/CSDC, links with CliMathNet, and through the planned 2019 workshop. The Innovation, Impact and Business (IIB) dept at Exeter will also facilitate engagement between the PI with relevant end-users. The PI/Co-I/PDRA will also engage with industry through study group events that utilize and develop cutting edge mathematical and statistical methods.

Impact will be delivered internationally through interactions with the VFs, their associated IIB departments and industrial contacts, and through external collaborators linked to CSDC and XCS. The (VF) Prof. Nicol works on research applications including extremes, coupled systems and recently in health care. The (VF) Dr Sterk works in dynamical systems, climate variability and prediction of extreme events and has collaborative links to the KMNI (Netherlands weather agency).

Through training of the PDRA this research will develop potential future leaders in dynamical systems and ergodic theory, and more broadly in scientific research where novel mathematical and statistical methods are required, such as in climate and weather prediction. Through outreach and public engagement impact will be delivered via presentations at science festivals, school masterclasses, and public lectures. The PI has extensive experience in this respect, and the PDRA will receive relevant training to deliver broader impact in their own research.