Mapping and visualization of singular surfaces in models for critical behaviour in excitable media

Supervisor

Dr. Rowena Ball Phone: 6125 2437

The emergence of abrupt changes, metamorphoses, oscillations, or “bad'' behaviour from smooth, continuous, or “good'' behaviour in continuum models of dynamical systems can be described predictively in terms of singularity theory and catastrophe theory. Often the onset of discontinuous action is ascribed to the propinquity of universal singularities such as fold, pitchfork, or Hopf bifurcations. In a 3-dimensional space defined by the system parameters, singular points may be unfolded into surfaces. A priori knowledge of the topology of these critical surfaces, and how they change with variation of other parameters (i.e., animation) is a powerful new tool for the design and optimization of dynamical systems and the control of jump, hysteresis, or oscillatory phenomena. In this Honours project one of the aims will be to calculate and visualize, and animate where appropriate, the bifurcation manifolds of selected dynamical models that exhibit discontinuous phenomena. A motivating theme is that understanding, controlling, and profiting from critical phenomena in statistical ensembles is vital to the development of new technologies for human development - nonlinear behaviour is one of the ``final frontiers''.

The project is envisaged as an extension of some recent work in singularity theory analysis of dynamical systems, in which bifurcation surfaces were calculated for (1) a thermally unstable reacting system, and (2) models for L(low)-H(high) confinement transitions in plasmas.

This project provides opportunities in the rapidly developing field of scientific visualization, with the particular application being novel 3-D visualizations of mathematical objects. It is expected that students will have access to the WEDGE, a new walk-in virtual environment being developed at RSPhysSE.

For more theoretically-minded students, this project provides scope to develop and apply the theory and mathematics of Poincaré, Whitney, Thom, Arnold, and others, to probe the relationship between mathematical models and the physical systems they purport to represent, and to help disseminate the methods of singularity theory to a broader community of scientists and engineers who are interested in real-world problems of discontinuous behaviour.