• Dynamical structures and self-organization in complex fluids.

• Dissipative and Hamiltonian dynamical systems.

• Singularity theory.

• Computer visualization of bifurcation surfaces.

• Dynamical models of transitions between low and high confinement states and associated oscillatory behaviour (edge-localized modes) in confined plasma systems.

• How bifurcations can sing to us.
• • Nonlinear dynamical models of real-world phenomena typically contain singular solutions, or bifurcation points. The pathology of first- and higher-order bifurcations can reveal much about the relationship between mathematical models and the physics of the systems they are supposed to represent. For example, if a higher-order bifurcation persists in a lower-order bifurcation set we begin to suspect that the model may be subtly over-determined, or that an undetected symmetry may be present. A higher-order bifurcation that is uncomfortably trapped in a lower-dimensional space may be ``sent home'' by appropriate perturbations. Consequently, the model is improved.The overall bifurcation structure of a dynamical model can thus ``sing'' the story - and the future - of an associated physical system.
  • Individual bifurcations also contain information. If we ``interrogate'' a bifurcation it will often ``sing'', or inform us of threats to the model's integrity, or provide clues to the origins of self-organizing behaviour.

• See Plasma Theory page.