Potential PhD topics

Pattern formation

Regular patterns, such as stripes, squares and hexagons, are ubiquitous in nature, and their formation and stability are governed by the intricate and complex interactions of symmetry and nonlinearity. Nonlinear interaction of waves in different directions can lead to the formation much more complicated and beautiful patterns: quasipatterns, spatio-temporal chaos and other forms of chaotic dynamics, depending on just how the waves interact. This project will involve using ideas from nonlinear dynamics: bifurcation theory, stability theory, three-wave interactions, chaos, symmetry and heteroclinic cycles, to understand the formation and stability of complex patterns such as quasipatterns, spatio-temporal chaos or turbulent spirals.

The distinct aspect of this project is that it will involve problems with two length scales, where waves of two different wavelengths can interact in many different ways. There will be emphasis on deep understanding of the underlying dynamics in the problem, using computational tools, bifurcation theory, asymptotic theory, weakly nonlinear theory, symbolic algebra, group theory, or whatever is needed. While the project will focus on solving a particular set of partial differential equations using asmptotic and numerical methods, one of the beauties of the nonlinear dynamics approach is that it can have wide applicability in different areas of mathematics, physics, chemistry or biology. The ideas that this project will explore have application to understanding patterns in fluid dynamics (the Faraday Wave experiment), soft matter physics (the formation of polymer quasicrystals) and chemistry (two-layer reaction-diffusion systems).

Relevant Recent Papers (complete list)

Nonlinear dynamics: symmetries and chaos

Global bifurcations are often responsible for creating chaotic dynamics in dissipative differential equations. Much of the complicated behaviour exhibited by chaotic systems can be explained by constructing maps (usually one-dimensional) that are valid near such a bifurcation. The presence of symmetry makes the analysis more difficult, and introduces the possibility of new types of phenomena: synchronisation, cycling chaos, and blow-out bifurcations. PhD projects would study these phenomena in cases where symmetry requires the use of higher-dimensional maps to describe the dynamics. One possible application is understanding spiral patterns in models of cyclic competitive behaviour

Relevant Recent Papers (complete list)

Please contact me if you'd like further information on potential PhD projects, or go to the School of Mathematics web page for details on how to apply for admission.

I encourage students to become independent thinkers as they develop a thesis, with the goal that the work be publishable. I enjoy self-motivated, thorough and dedicated students. In turn, you can expect my supervision to be structured and involved. I am prompt with answering questions and open to meet with students as the need arises.

A full list of PhD topics in the Department of Applied Mathematics can be found here.