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Research Interests

My general interests involve applying dynamical systems techniques to problems involving the emergence of fully localised patterns.

Localised patterns are characterised by one or more regions of a locally periodic structure (i.e. a pattern) surrounded by a quiescent state. Such structures have been observed in many settings: such as in fluid mechanics, desert vegetation, and nonlinear optics, to name a few.

 

While localised patterns are well-studied in one spatial dimension, our mathematical understanding of spatial localisation in higher dimensions remains limited.

The focus of my research has been on improving our understanding of localised patterns via a combination of analysis, modelling, and numerical simulations.

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"Core Far-field" decomposition of a solution localised in the spatial coordinate r
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Extended phase space between the core and far-field manifolds, with leading-order solutions (blue arrows)

Localised Patterns on a Ferrofluid

Ferrofluids consist of magnetic nanoparticles suspended in a carrier fluid, forming a magnetic fluid. There are many interesting experiments involving ferrofluids, including the `Rosensweig’ experiment where a dish of ferrofluid is placed between two Helmholtz coils emitting a vertical magnetic field. For a strong enough magnetic field, spikes begin to appear on the surface the ferrofluid (see figure).

My interest lies in studying the existence and stability of localised axisymmetric structures on the surface of a ferrofluid. This problem involves significantly complicated equations and requires the formulation of new theory for radial quasilinear PDE systems. However, it does allow for strong collaboration with experimental projects such as those carried out by Reinhard Richter et al. at the University of Bayreuth.

I am also interested in extending these ideas to other applications of localised radial patterns, such as in reaction-diffusion models for desert vegetation and non-local models for neural networks.

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Photo of the experiment, where a localised radial spot emerges from the ferrofluid
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Multiple spikes can emerge from the ferrofluid 

Localised Dihedral Patterns

Localised patches with dihedral symmetry–such that they are invariant under discrete rotations–have been observed in soft matter, nonlocal models, and various experiments. However, structures with localisation in more than one direction are notoriously difficult to analyse via dynamical systems.

At this point, many questions remain regarding localised dihedral patterns. Do such structures exist mathematically? Are they ever stable? How many different types of localised cellular patterns are there?

I am very interested in developing our theoretical understanding of these localised dihedral patterns, including their existence, stability, and bifurcation structure. I would also like to better understand localised cellular patches in models where such structures have been observed, such as the ferrofluid experiment (see figures), nonlinear optics, soft matter, and many more.

 

As part of this work, I employ numerical continuation methods to find localised dihedral patterns in the Swift-Hohenberg equation and map out their bifurcation diagrams in parameter space.

 

 

Videos are available on the Videos page, highlighting some localised dihedral patterns that have been found from this approach. Relevant MATLAB codes can be found here

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Localised hexagon found from numerical simulations
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Weakly localised patterns with various dihedral symmetries
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Examples of localised hexagonal rings
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