Nonlinear Dynamics and Asymptotics

Mixed-mode oscillations in singularly perturbed three-timescale systems

Supervisor(s): Nikola Popovic

Many natural systems are characterised by the presence of multiple timescales. Considering the whole Universe itself, for instance, stellar phenomena are very slow compared to Earthly ones, for example compared to how long it takes for Earth to complete a rotation around the Sun or to how long it takes for Earth to complete a rotation around its axis. The timescale-separation is even more extreme when such macroscopic phenomena are compared to microscopic ones, like atomic and subatomic phenomena etc.

However, different timescales are apparent even in smaller, more contained and maybe closed systems, were the dynamics between different components are not as seemingly unrelated as in the example of Earth’s orbit and atomic particles. Some examples of such systems include biological systems, were electrical phenomena happen faster than chemical ones and where even chemical reactions can occur at very different rates, or climate systems, where pressure, temperature, humidity and other quantities vary at different rates, leading to complicated meteorological phenomena, like the weather in Scotland. The focus of this work is the study of a class of dynamical systems, i.e. of mathematical formulations that are used to model natural systems, that are characterised by the presence and interaction of three distinct timescales. The aim is to uncover the mechanisms that distinguish between qualitatively different oscillatory dynamics of the system, explaining, for instance, why for some particular choices of the parameters of the system only two timescales seem to manifest themselves, while for other choices of the parameters, all three timescales seem to manifest themselves in the behaviour of the system.

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Geometric singular perturbation theory for reaction-diffusion systems

Supervisor(s): Nikola Popovic, Mariya Ptahsnyk

The indispensability of mathematical multiple scale theory across a large breadth of scientific fields cannot be overstated. Almost every physical process exhibits dynamics in varying scales of length, time or energy and may involve highly non-trivial interactions between quantities evolving on these different scales, necessitating the development of rigorous mathematical theories and methods that are able to qualitatively explain and predict the behaviour of such processes by exploiting the underlying multiple scale structure. Many multiple scale models also have a spatial dependence, that current approaches are not able to handle properly due to immense mathematical complexities this presents. This results to simplifying assumptions such as averaging over the spatial domain in order to tackle the spatial dependence. The contribution of this work is the development of novel methods that allow dealing with this complexity in the original, unsimplified problem.

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Rigorous asymptotics for the Lamé, Mathieu and spheroidal wave equations with a large parameter

Supervisor(s): Adri Olde Daalhuis

Real world phenomena are frequently described in science by differential equations. Since their introduction by Newton and Leibniz in the latter half of the 17th century, the theory of such equations has been extensively studied by generations of mathematicians and physicists. Many differential equations have a free parameter. When this parameter takes on special values, termed eigenvalues, the di↵erential equation admits special solutions called eigenfunctions. These are the solutions which are typically of interest in physical applications. Often differential equations cannot be solved explicitly, and thus some approximation theory is needed to describe solutions, and in turn the corresponding eigenvalues, if they exist. If the case of interest concerns some variable or parameter which tends to some limit, we do this using asymptotic methods. This thesis concerns three di↵erential equations, the so-called Lam´e, Mathieu and spheroidal wave equations. The case of interest in physical applications is a parameter in these equations becomes large. We concentrate on approximating the eigenfunctions and eigenvalues in each of these three cases, using uniform asymptotic methods.

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Nonlinear waves In nematic liquid crystals

Supervisor(s): Noel Smyth, Tom Mackay

Optics is the branch of physics that studies the behaviour and properties of light and its interaction with matter. Commonly, visible light refers to electromagnetic radiation that the human eye can perceive, but in Physics, the term light refers to electromagnetic radiation of any wavelength, whether visible or not. The experimental development of laser beams (light sources that produce a very narrow beam of a single wavelength or colour) in 1960, instigated the interest in new materials and their interaction with powerful optical beams. One of these fascinating materials is liquid crystals, chemical compounds whose properties are between fluid and solid crystallines, they are characterized for interacting strongly with electromagnetic fields (in particular, laser beams).

Liquid crystals offer a wide range of technological and scientific applications, currently being used in devices such as smart screens, mobile phones and televi- sions. Moreover, their versatility can play a role in future optical communications, signal processing and computational progress. This thesis studies the interaction between laser beams and a particular type of liquid crystal material, examining the path followed by the light as it travels along the material and the potential design of an optical isolator, a device that distinguishes light signals travelling in one direction from those travelling in the opposite one.

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