Oxford Centre for Industrial and Applied Mathematics

Dr Leonard A Smith

Senior Research Fellow, Pembroke College

Research Interests

My primary interests lie in understanding the dynamics of nonlinear systems, both real and abstract; I am particularly interested in developing methods which are robust in the face of real data, and allow the modelling an optimal prediction of both chaotic and stochastic nonlinear systems.

Nonlinear systems are commonly characterised by examining their numerical representations. For chaotic systems per se, we may estimate Lyapunov exponents, dimensions, and unstable periodic orbits; I am interested in establishing the reliability of such estimates, distinguishing fundamental dynamics from numerical artifact, and determining the implications these quantities hold for prediction and control. How might we identify chaos in physical systems?

When considering data from physical experiments, we are concerned with the ubiquitous rather than the generic. A number of problems arise in such systems, including the effects of observational uncertainty, dynamical noise, quantisation and finite length data records. The best method for dealing with these effects depends on the application (diagnostic or prognostic) and even the particulars of the system. In many complicated systems - like the IPCC record of the Earth's mean global temperature - the available data is insufficient to make detailed diagnostics of the dynamics, yet we can evaluate the likelihood of observed "oscillations" being indicative of determinism, or merely arising by chance. I have applied similar techniques to detect sensor failure of a change in the dynamics of complex mechanical systems without requiring a model of the system a priori.

My central interest is in the prediction problem: determining the relevant measurements in a data rich environment, constructing nonlinear models and evaluating the results. One project simulates weather forecasting in the laboratory through the study of a rotating, thermally forced fluid annulus, another project evaluates the growth of uncertainty in financial markets in real time. While the construction of nonlinear models is now common, linear intuitions applied to their interpretation can be misleading. I am currently applying an ensemble prediction approach to a number of systems including laser instabilities, short range surface temperature forecasts, medium term climatic forecasts, computational ecosystems, solar physics, and impact systems, in addition to the annulus experiments. This approach not only quantifies the quality of predictions as they are made, but also allows us to determine whether additional resources are best invested in improving the quality of the input data, constructing a better model, or obtaining additional computational power.

Papers to download

Contact Information

Tel: 01865-2-70517
E-mail: lenny@maths.ox.ac.uk

Last updated: June 1996

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