This PhD thesis is devoted to the study of dynamical systems appearing in theoretical models of dark energy. The quest for understanding the origin of the observed cosmic acceleration has led physicists to advance a large number of phenomenological explanations based on different fundamental theories. The best approach to analyse the background cosmological impli- cations of all these models consists in employing dynamical systems tech- niques. In this thesis, after reviewing elements of dynamical systems theory and basic cosmology, several dynamical systems, which arise in dark energy models ranging from scalar fields to modified gravity, will be studied using both analytical and numerical methods. The work is organised in order to present as many details as possible for the simpler and well known models, while outlining major results and referring to the literature for the less stud- ied ones. This choice aims at providing the reader with a complete overview and summary of dynamical systems in dark energy applications.

This thesis concerns operators whose Weyl pseudodifferential operator symbol is the convolution of a function that is smooth and of fixed scale with a function that is discontinuous and dilated by a large asymptotic parameter. A special case of these operators of particular interest is the class of generalised anti-Wick operators, and then the fixed-scale part corresponds to the window functions while the dilated part is the generalised anti-Wick symbol. The main result is a Szegő theorem that gives two terms in the asymptotic expansion of the trace of a function of the operator. Two variants are proved: in one the discontinuity must occur on a C^2 surface but the symbol may have unbounded support, while in the other the set on which the discontinuity occurs may be much more general (most importantly, it must be Lipschitz and piecewise C^2), but the symbol must be compactly supported. A corollary of this theorem is two terms in the asymptotic expansion of the eigenvalue counting function when the smooth part of the symbol is constant. Prior to this work, only one term in each of these expansions was known. It is also shown that the remainder in the Szegő theorem is larger for a class of examples where the boundary has a cusp; this shows that the Lipschitz condition in the main theorem cannot be removed without weakening the conclusion. A significant step in the proof of this Szegő theorem is a composition result for Weyl pseudodifferential operators that may be of more general interest: the symbol of the composition is expressed as a finite series in the standard form, but with an explicit trace norm and operator norm bound of the remainder expressed using the symbols in a similar way to the first excluded term. In the one-term case, this is used to derive an analogous trace norm bound for approximating the Weyl symbol of a function of an operator. Another important part of the proof of the Szegő theorem is the use of standard tubular neighbourhood theory to describe the geometry of the surface on which the discontinuity occurs; this is derived in full for the necessary conditions.

In this dissertation the structure of singularities in the complex plane of solutions of certain classes of ordinary differential equations and systems of equations is studied. The thesis treats two different aspects of this topic. Firstly, we introduce the concept of movable singularities for first and second-order ordinary differential equations. On the one hand the local behaviour of solutions about their movable singularities is investigated. It is shown, for the classes of equations considered, that all movable singularities of all solutions are either poles or algebraic branch points. That means locally, about any movable singularity z0, the solutions are finitely branched and represented by a convergent Laurent series expansion in a fractional power of z-z0 with nite principle part. This is a generalisation of the Painleve property under which all solutions have to be single-valued about all their movable singularities. The second aspect treated in the thesis deals with the global structure of the solutions. In general, the solutions of the equations discussed in the first part have a complicated global behaviour as they will have infinitely many branches. In the second part conditions are discussed for certain equations under the existence of solutions that are globally nite-branched, leading to the notion of algebroid solutions. In order to do so, some concepts from Nevanlinna theory, the value-distribution theory of meromorphic functions and its extension to algebroid functions are introduced. Then, firstly, Malmquist's theorem for first-order rational equations with algebroid solutions is reviewed. Secondly, certain second-order equations are considered and it is examined to what types of equations they can be reduced under the existence of an admissible algebroid solution.

The distribution of incidents in time and space is a central issue in the study of crime, for both theoretical and practical reasons. It is also a context in which quantitative analysis and modelling has significant potential value: such research represents a means by which the implications of theory can be examined rigorously, and can also provide tools which support both policing and policy-making. The nature of the field, however, presents a number of challenges, particularly with regard to the incorporation of complex environmental factors and the modelling of individual-level behaviour. In this thesis, the techniques of complexity science are used to overcome these issues, and the approach is demonstrated using a number of examples from a range of crime types. The thesis begins by presenting a network-based framework for the analysis of spatio-temporal clustering. It is demonstrated that signature `motifs' can be identified in patterns of offending for burglary and maritime piracy, and that the technique provides a more nuanced characterisation of clustering than existing approaches. Analysis is then presented of the relationship between street network structure and the distribution of urban crime. It is shown that burglary risk is predicted by the graph-theoretic properties of street segments; in particular, those which correspond to levels of street usage. It is further demonstrated that the `near-repeat' phenomenon in burglary displays a form of directionality, which can be reconciled with a novel street network metric. These results are then used to inform a mathematical model of burglary, which is situated on a network and which may be used for prediction. This model is analysed and its behaviour characterised in terms of urban form. Finally, a model is presented for a contrasting crime problem, the London riots of 2011, and used to examine a number of policy questions.

A general treatment of the dynamics and statistical mechanics of point vortices in bounded domains is introduced in Chapter 1. Chapter 2 then considers high positive energy statistical mechanics of 2D Euler vortices. In this case, the most-probable equilibrium dynamics are given by solutions of the sinh-Poisson equation and a particular heart-shaped domain is found in which below a critical energy the solution has a dipolar structure and above it a monopolar structure. Sinh-Poisson predictions are compared to long-time averages of dynamical simulations of the $N$ vortex system in the same domain. Chapter 3 introduces a new algorithm (VOR-MFS) for the solution of generalised point vortex dynamics in an arbitrary domain. The algorithm only requires knowledge of the free-space Green's function and utilises the exponentially convergent method of fundamental solutions to obtain an approximation to the vortex Hamiltonian by solution of an appropriate boundary value problem. A number of test cases are presented, including quasi-geostrophic shallow water (QGSW) point vortex motion (governed by a Bessel function). Chapter 4 concerns low energy (positive and negative) statistical mechanics of QGSW vortices in `Neumann oval' domains. In this case, the `vorticity fluctuation equation' -- analogous to the sinh-Poisson equation -- is derived and solved to give expressions for key thermodynamic quantities. These theoretical expressions are compared with results from direct sampling of the microcanonical ensemble, using VOR-MFS to calculate the energy of the QGSW system. Chapter 5 considers the distribution of 2D Euler vortices in a Neumann oval. At high energies, vortices of one sign cluster in one lobe of the domain and vortices of the other sign cluster in the other lobe. For long-time simulations, these clusters are found to switch lobes. This behaviour is verified using results from the microcanonical ensemble.

Mathematical modelling of civil violence can be accomplished in different ways. In this thesis, four modelling frameworks are investigated, each of which leads to different insights into the spatio-temporal properties of civil violence. These frameworks vary with respect to the extent in which empirical data is used in generating model assumptions, and the extent in which simplifying assumptions distance the model from the real world. An overarching objective is to compare the insights and underlying assumptions of each framework, and to consider how they might be consolidated to aid policy decision-making. Within each framework, novel contributions both to the mathematical modelling of social systems, and to the theoretical understanding of civil violence are made. First, a novel data-driven approach for analysing local patterns of geographic diffusion in event data is presented, and applied to offences associated with the 2011 London riots. Second, by considering the decision-making of individuals, thereby taking an agent-based perspective, and using existing theory to construct model assumptions, a parametric statistical model of discrete choice is derived that more closely inspects the targets chosen by rioters, and how these choices might have changed over time. The application of this model to the policy domain is explored by considering police deployment strategies. Third, focusing on the interaction between two adversaries, and employing stochastic point process models, a series of multivariate and nonlinear Hawkes processes are proposed and used to explore spatio-temporal dependency during the Naxal insurgency in India. Fourth, a novel spatially-explicit differential equation-based model of conflict escalation between two adversaries is derived. A bifurcation is identified that results from the spatial disaggregation of the model. Implications for the interpretation of the model in the real world and potential applications are discussed.

It is well-known that the probabilistic behaviour of financial asset returns is not captured well by the classical Black-Scholes model. The true behaviour will never be perfectly captured in any model, but insight is continually being obtained into our understanding of more sophisticated and realistic models. Much research has been published recently exploring the use of L\'vy process models, which maintain the original \emph assumption present in the Black-Scholes model, but incorporate jumps in the modelling. This investigation seeks to motivate a new class of models, throwing out the stationary increments hypothesis. We argue that certain techniques of trading decision-making are not independent of previous price movements, and the returns, being driven by the trade order flow, will reflect that. From here, we develop two particular such models, which are both diffusion models, and study them for their probabilistic behaviour. The first of these models is a hybrid of the arithmetic and geometric Brownian motions, which has transition probabilities expressible in terms of a spectral expansion involving Legendre functions. The second is a hybrid of the arithmetic Brownian motion and the Cox-Ingersoll-Ross process, and its spectral expansions involve the confluent hypergeometric functions. Having developed these expressions in sufficient detail to do so, we consider the calculation of value-at-risk and expected shortfall in these two models.

We generalise the notion of δ-scattered to partial orders and prove that some large classes of δ-scattered partial orders are better-quasi-ordered under embeddability. This generalises theorems of Laver, Corominas and Thomassé regarding δ-scattered linear orders, δ-scattered trees, countable pseudo-trees and N-free partial orders. In particular, a class of countable partial orders is better-quasi-ordered whenever the class of indecomposable subsets of its members satisfies a natural strengthening of better-quasi-order. We prove that some natural classes of structured δ-scattered pseudo-trees are better quasi-ordered, strengthening similar results of Kříž, Corominas and Laver. We then use this theorem to prove that some large classes of graphs are better-quasi-ordered under the induced subgraph relation, thus generalising results of Damaschke and Thomassé. We investigate abstract better-quasi-orders by modifying the normal definition of better-quasi-order to use an alternative Ramsey space rather than exclusively the Ellentuck space as is usual. We classify the possible notions of well-quasi-order that can arise by generalising in this way, before proving that the corresponding notion of better-quasi-order is closed under taking iterated power sets, as happens in the usual case. We consider Shelah's notion of better-quasi-orders for uncountable cardinals, and prove that the corresponding modification of his definition using fronts instead of barriers is equivalent. This gives rise to a natural version of Simpson's definition of better-quasi-order for uncountable cardinals, even in the absence of any Ramsey-theoretic results. We give a classification of the fronts on [K]^ω, providing a description of how far away a front is from being a barrier.

In this thesis analytical and numerical methods are used to investigate the stability and propagation of waves in various fluid-filled elastic-walled channels. In Chapter 2 a linear temporal stability analysis is performed for various channel wall configurations, the results of which are compared to determine how different physical effects contribute to the stability. In Chapter 3 a linear spatial stability analysis is performed for flow between linear plates, this is a limiting case of the wall configuration in Chapter 2. Roots of the dispersion relation are found and their stability and direction of propagation determined. Chapters 4 and 5 focus on the propagation of nonlinear waves on liquid sheets between thin infinite elastic plates. Both linear and nonlinear models are used for the elastic plates. One-dimensional time-dependent equations are derived based on a long wavelength approximation. Symmetric and antisymmetric travelling waves are found with the linear plate model and symmetric travelling waves are found for the fully nonlinear case. Numerical simulations are employed to study the evolution in time of initial disturbances and to compare the different models used. Nonlinear effects are found to decrease the travelling wave speed compared with linear models. At sufficiently large amplitude of initial disturbances, higher order temporal oscillations induced by nonlinearity can lead to thickness of the liquid sheet approaching zero. Finally, in Chapter 6, an extension of the model from Chapters 4 and 5 is considered, the elasticities of the two channel walls are allowed to differ. The effect of this difference is determined through a mix of techniques and the limiting case of one wall having no elasticity is discussed.

If we have topological conjugacy between two continuous maps, T : X → X and T 0 : X0 → X0 , then counts of closed orbits and periodic points are preserved. However, if we only have topological semi-conjugacy between T and T 0 , then anything is possible, and there is, in general, no relationship between closed orbits (or periodic points) of T and T 0 . However, if we let a finite group G act on X, where the action of G commutes with T and where we let X0 = G\X be the quotient of the action, then it is indeed possible to say a bit more about the relationship between the count of closed orbits of (X, T) and its quotient system (X0 , T0 ). In this thesis, we will describe the behaviour of closed orbits in quotient systems, and we will show that there exists a wide but restricted range of what growth rates can be achieved for these orbits. Moreover, we will examine the analytic properties of the dynamical zeta function in quotient systems.