POSTPONED
to later in the year

Dynamical systems and its attractors
This course will cover the modern theory of dynamical systems by discussing the basic phenomena. It will draw together both geometric, analytical and computational approaches to dynamical systems.

Topics to be covered: Elementary attractors of ode and discrete systems; examples of global attractors; chaos and sensitive dependence on initial conditions; statistical analysis of deterministic dynamical systems.

Modelling
A range of models from the physical sciences, business and biology will be discussed during the course. These will be chosen to illustrate general approaches to the development and analysis of models, but some attempt will be made to adapt them to the specific interests of the audience.

Time Series Analysis
This course will cover both the new nonlinear systems approaches to time series analysis, as well as associated statistical techniques.

Topics to be covered: Power spectra: singular value decomposition, fractal and correlation dimensions, modelling chaotic time series. Nonlinear filtering techniques: separation of noise and chaos, use of nonlinear noise models, wavelet transformations.

Perturbation theory, Asymptotics and Pattern Formation
The goal of the course is to explore ways of exploiting the presence of small parameters in order to gain insight and understanding through analytical methods of the behaviour of complicated systems. This theoretical approach has led to important advances in many areas.

Topics to be covered: Elementary boundary layer theory, approximating solutions to such boundary value problems. Coupled nonlinear oscillators, averaging, multiple scales, resonance, phase locking, synchronisation. Behaviour of damped and driven systems near bifurcations and phase transitions, asymptotic analysis of integrals and the method of stepcut descent order parameters, Landau equations instabilities and patterns.

Computation in ODEs and PDEs
This course will introduce the latest numerical techniques for the integration of ordinary and partial differential equations. The emphasis will be on practical techniques, and participants will participate in practical computer based workshops.

Topics to be covered: Numerical methods of simulation and bifurcation analysis for nonlinear systems, including both odes and pdes. Methods for characterising chaos: fractal dimensions and Lyapunov exponents. Visualisation and interactive computation of nonlinear systems, including the most recent software packages: Matlab, Dstool, Auto98

Bifurcation theory
In this course we study the ways in which nonlinear systems change their qualitative behaviour as a parameter is varied. Modern methods allow one to do this for complex systems and it is one of the most effective tools in the analysis of nonlinear systems.

Topics to be covered: Local bifurcations: bifurcations of equilibria, Hopf bifurcation, period doubling, exchange of stability, effect of symmetry, centre manifold reduction and normal forms, global bifurcations: heteroclinic bifurcations, bifurcations of attractors.