Björn Sandstede
Björn Sandstede
The formation of patterns and waves has long been of interest across the physical and life sciences. From the stripes of a zebra and the spots on a leopard's back to the ripples on a sandy beach or desert dune, regular spatially extended patterns arise everywhere in nature. Spatially localised patterns like sunspots, beginnings of tumours or single convection cells in heated liquids also occur frequently. Both localised and extended patterns, and their dynamics, are investigated here at Surrey in the Dynamics of Patterns research group.
The following projects are available to PhD students. This is not an exhaustive list, and we are happy to discuss and formulate other projects.
When two different spatially periodic patterns collide, an interface may form between them which we refer to as a defect. An example is
where the left picture shows a space-time plot of a defect (colours indicate different values of the solution) and the right picture is a graph of the solution for a fixed time. Defects can, for instance, arise in bifurcations from localised pulses as bound states between pulses and periodic patterns: An interesting question is to analyse the interaction between the pulse and the periodic patterns if the latter are slightly shifted in space. Open are also extensions to multi-dimensional problems, for instance, the analogous bifurcation of planar target patterns from localised radially symmetric structures.
Spiral waves have been found in many experiments such as the famous Belousov-Zhabotinsky reaction and the oxidation of carbon-monoxide on platinum surfaces. In the former reaction, period doubling of spiral waves has been observed:
This bifurcation has been analysed in 2D. Of interest would be an analytic and numerical study of this phenomenon in 3D where the line defect, which connects the core to the boundary in the picture above, becomes a two-dimensional surface which itself may exhibit a interested dynamics.
In recent experimental work, Reinhard Richter (University of Bayreuth) found localised patterns with ferrofluids which change their properties when exposed to a magnetic field:
Of interest to experimentalists are questions like what type of patterns can be observed? When do they exist? And when are they stable? Projects in this area may look at phenomenological modelling of ferrofluid-experiments, numerical methods for 2D/3D localised patterns in discrete/continuous media or analysis of coherent structures.