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My main interests are in the study of pattern formation and in using ideas from dynamical systems theory to understand the behaviour of various physical and biological systems.
Regular patterns are seen in a wide variety of physical, chemical and biological systems. The tools of symmetry and bifurcation theory can be used to determined some of the generic features of such patterns and which ones are most likely to be observed. One of the focusses of my research is the study of superlattice patterns initially understanding the generic bifurcation problem and more latterly applying these results to specific physical problem. Some examples of superlattice patterns are here .
One problem of particular interest is understanding the huge variety of patterns that arise in the Faraday crispation experiment. In this experiment a container of fluid is shaken up and down. Depending on the frequency and amplitude of the shaking different patterns are observed. My current research on this problem includes work with Alastair Rucklidge and Mary Silber on a mechanism for the development of spatio-temporal chaos.
Many mathematical studies of patterns assume that the domain is infinite. In practice, experiments are necessarily finite and boundaries can have an influence on the patterns that are observed. With Peter Daniels and Diep Ho I have been examining how boundaries can influence the patterns that are observed in convection problems.
Mathematical applications
Models of population dynamics including delay. (Collaborator: Nicholas Robertson).
Modelling the oxygenation of tumours and PET tracer dynamics. (Collaborators: David Lloyd, Andrew Nisbet, David Bradley, Entesar Dalah)
I am a co-investigator on the Evolution and Resilience of Industrial Ecosystems project, ERIE .
Novel spiralling carbon nanostructures. (Collaborators: David Lloyd, Hidetsuga Shiozawa, Vlad Stolojan, David Cox and Ravi Silva).
The dynamics of the carbon cycle of an evergreen forest. (Collaborators: Anna Chuter, Philip Aston).
The dynamics of the sleep-wake cycle. (Collaborators: Gianne Derks, Derk-Jan Dijk).
Past projects include:
Invesigating a a mathematical model of the nephron, the key functioning unit of a kidney. (Collaborator: Ian Purvey)
Dynamical systems: bifurcations and chaos
Since the discovery of chaotic phenomena, it is well-known that simple systems of ordinary differential equations can display extremely complex dynamics. In my thesis I examined one such simple system, a system of parametrically excited coupled pendulums, in order to investigate the routes to chaos.
Geological evidence tells us that the magnetic field within the Earth has intermittently changed its direction, with magnetic North becoming South and vice versa. With Raymond Hide and David Acheson I investigated simple models of geodynamos to illustrate this phenomena.