Ph.D. Research Projects with John Hogan

Graph theory and dynamical systems of arbitrary dimension.

A lot is now known about the nonlinear dynamics of systems of low dimension. For example the use of Poincare sections, Lyapunov exponents and bifurcation techniques are well established in this field. However in systems of large dimension, very little is known indeed. Recently, graph theory and percolation theory have been seen as providing a means to basic discoveries in systems of large dimension. This project will build on recent work with directed graphs, to apply the theory of random graphs to estimate the effective lifetime of a heat exchanger. No knowledge of graph theory is needed.

Disclinations in liquid crystals.

Liquid crystals provide some of the most difficult problems to confront the applied mathematician, whilst at the same time highlighting links with other fundamental problems in mechanics, such as turbulence & fracture. One of the most important problems concerns the effect of imperfections (known as disclinations) in the liquid crystal make-up. Great scope exists for both numerical and theoretical breakthroughs in this area. No knowledge of liquid crystals is needed to start the project. Excellent links exist with Sharp Laboratories of Europe, the research arm of the world's largest liquid crystal manufacturer, who can be expected to provide an excellent set of problems if required. There is a very strong possibility that this studentship will be converted into a CASE award.

Piecewise smooth systems.

Bristol has an outstanding reputation for work in this area. These systems contain impacts or changes in ambient conditions such as control. They appear in thousands of applications from earthquake engineering to the mechanics of the inner ear. Several different problems are available for the enterprising student, ranging from the very theoretical (deciding just how a system does behave near impact) to completely numerical (how to numerically continue these solutions) or even experimental. Just ask!

Chaotic communications.

The Department has excellent links with DERA in Malvern who already sponsor our work on chaotic communications. This project will involve exploration of some of the mathematical models which lie behind some of the proposed new algorithms in this area. Some knowledge of dynamical systems theory would be useful.

Swimming pools & potable water

Following development of a simple model for swimming pool chlorination sponsored by the Danish company, Grundfos, members of the ANM group have begun a collaboration with Cranfield University to expand the model & compare with experiments. This project will build on this work by moving the modelling into the areas of water purification (applications to potable water). An understanding of mathematical modelling & a willingness to be immersed in chemistry is useful.

Mathematical modelling of emphysema.

Emphysema is a terrible lung disease which strikes smokers or those who work in dusty environments. This project is the next stage in a long-standing & very successful collaboration between the ANM group & the Bristol Royal Infirmary (BRI). Using images from the BRI medical cameras, we plan to develop an existing dynamical model to include spatially extended effects. A mathematical project with healthy benefits!