# Extreme events in deterministic systems

## Prof Dr Henk W Broer and Dr Alef E Sterk from the University of Groningen discuss how mathematical modelling offers a fruitful approach towards understanding climatic extremes…

Dynamical systems are mathematical models for everything that evolves in time. For example springs and pendulum clocks. More complicated examples are the atmosphere that produces the everyday weather, and the celestial bodies comprising the solar system. These systems are deterministic in the sense that the present state of the system completely determines its future. Time series produced by deterministic systems often look random, and probabilistic tools can be very useful in studying deterministic systems.

Since the seminal work of the mathematician and meteorologist E.N. Lorenz in the 1960s it is well known that deterministic systems can be very unpredictable: small errors in the initial state may lead to large errors in later states. This phenomenon, which is colloquially known as chaos, hampers long-term weather forecasts and stimulated the development of mathematical research on nonlinear dynamics and chaos theory. A recent development in the theory of deterministic dynamical systems is the study of extreme value statistics. This is particularly useful for applications in weather and climate.

Mathematical modelling offers a fruitful approach towards an understanding of meteo-climatic extremes. Models for the atmospheric and oceanic circulation are typically derived from first principles, such as Newton’s laws, conservation of energy, global balances, etc. This approach leads to a set of equations describing the evolution of quantities like pressure, temperature, and wind speed. Often these equations can only be solved numerically using high performance computers. deterministic, its evolutions can be studied using a combination of geometric and statistical techniques.

The statistics of large values in a time series are described by the generalised extreme value distribution. A particularly important parameter of this distribution is the so-called tail index because it determines the tail width of the distribution and therefore the frequency and intensity of extreme events. The tail index is often estimated by running long simulations and applying statistical inference methods. It is still an open question how much data is needed in order to obtain an accurate estimate. Research has shown that already for very simple models the computation of the tail index requires prohibitively long time series. If this is also the case for state of the art climate models, then quantifying the extreme behaviour in such models might be a serious problem.

Studying the statistics of extremes in climate models is an interdisciplinary problem and requires the joint efforts of both climate scientists and pure mathematicians. Future research could be aimed at developing novel techniques for estimating the tail index that do not rely on computing long time series.

Prof Dr Henk W Broer

Professor of Dynamical Systems

Tel: +31 50 363 3959

h.w.broer@rug.nl

www.math.rug.nl/dsmp/People/HenkBroer

Dr Alef E Sterk

Assistant Professor

Tel: +31 50 363 3975

a.e.sterk@rug.nl

Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen

www.math.rug.nl/dsmp/People/AlefSterk