In this PhD thesis, the role of dynamical systems in cosmology has been studied. Many systems and processes of cosmological interest can be modelled as dynamical systems. Motivated by the concept of hypothetical dark energy that is believed to be responsible for the recently discovered accelerated expansion of the universe, various dynamical dark energy models coupled to dark matter have been investigated using a dynamical systems approach. The models investigated include quintessence, three-form and phantom fields, interacting with dark matter in different forms. The properties of these models range from mathematically simple ones to those with better physical motivation and justification. It was often encountered that linear stability theory fails to reveal behaviour of the dynamical systems. As part of this PhD programme, other techniques such as application of the centre manifold theory, construction of Lyapunov functions were considered. Applications of these so-called methods of non-linear stability theory were applied to cosmological models. Aforementioned techniques are powerful tools that have direct applications not only in applied mathematics, theoretical physics and engineering, but also in finance, economics, theoretical immunology, neuroscience and many more. One of the main aims of this thesis is to bridge the gap between dynamical systems theory, an area of applied mathematics, and cosmology, an exciting area of physics that studies the universe as a whole.

In this thesis we develop a new criterion for the congruence subgroup problem in the case of arithmetic groups of $\SU(2,1)$, which in principle can be checked using a computer. Our main theorem states that if there exists a prime $q>3$ and a congruence subgroup $\Gamma'\subset \SU(2,1)(\Z)$ such that the restriction map $H^{2}(\SU(2,1)(\Z), \F_{q}) \rightarrow H^{2}(\Gamma',\F_{q})$ is not injective, then the congruence kernel of $\SU(2,1)$ is infinite.

Variations in shelf geometry mean that a coastal-trapped wave (CTW) mode can propagate within some finite length of shelf but be evanescent outside these regions. These disturbances will be denoted here as localised coastal-trapped waves (ℓCTWs). The subsequent motions produced by such localised variations in shelf geometry, i.e. wave propagation, wave trapping and eddy generation, are discussed in both the coastal-trapped wave (baroclinic) and continental shelf wave (barotropic) limits. Firstly, localised continental shelf waves (ℓCSWs) trapped by local variations in coastline curvature, shelf slope and shelf break distance from the coastal wall are constructed using a WKBJ approximation (accurate for both strong and weak variations) and an expansion about cut-off frequencies (valid for sufficiently weak variations). Comparison with a full numerical study, based on spectral differentiation matrices, of the nonlinear differential eigenvalue problem demonstrates that, in their required limits, both approximations are extremely accurate. The numerical scheme is also extended to discuss how these localised disturbances may be generated by localised wind forcing oscillating at a frequency close to that of the localised modes thereby generating a resonant response in the neighbourhood of the local geometric variation. The second body of work describes efficient and accurate spectral numerical schemes to compute both propagating and evanescent free baroclinic coastal-trapped waves over general depth profiles for arbitrary density profiles in horizontally semi-infinite domains. A novel nonlinear boundary condition is derived that is particularly effective for modes whose offshore decay is weak, as in the long-wave limit. Additionally, geographically localised coastal trapped waves are constructed both asymptotically, using a WKBJ approach, and numerically, using a 3D extension to the highly accurate 2D spectral schemes mentioned above, which allows for the inclusion of arbitrary alongshore variations in offshore depth profile and arbitrary vertical density profiles. Both schemes are then used to demonstrate the importance of stratification, shelf slope and shelfbreak distance from the coastal wall on ℓCTWs. Finally, a new mechanism for vortex generation on continental shelf margins is proposed by considering an incident CTW impinging a geographically localised non-propagating region. The nonlinear governing equations are integrated numerically on a spectral grid with the results used to discuss the interaction of nonlinearity, dispersion and viscosity, and thus the possibility of vortex generation, wave reflection or dissipation, as the incident wave mode approaches the singular region.

Baroclinic instability, the fundamental mechanism underlying the generation of baroclinic eddies in the atmosphere and ocean is investigated in the two-layer, quasi-geostrophic model. The aim is to bridge the gap in understanding between analytical theories and high resolution numerical simulations of more realistic flows. In chapter 1 the physical motivation for the problems, two-layer model and numerical scheme are introduced. In chapter 2, the instability of a uniform flow profile without Ekman friction is investigated. The success of a weakly nonlinear theory due to Warn & Gauthier at finite criticality is assessed over the full parameter space. The relevance of nonlinear bounds on wave amplitude and perturbation energy due to Shepherd is also evaluated. Chapters 3 and 4 investigate the Holopainen instability, whereby a uniform flow profile, otherwise stable in frictionless flow, is destabilized by the addition of a small amount of Ekman friction. In chapter 3, the physical mechanisms of the baroclinic and Holopainen instabilities are contrasted in terms of potential vorticity disturbances. The instability of the Eady model is also discussed. In chapter 4, a weakly nonlinear theory due to Romea is shown to be accurate for flows unstable to the Holopainen instability and flows unstable to baroclinic instability in the presence of significant Ekman friction. An intermediate flow region is found where Warn & Gauthier’s theory is accurate at early times, but the final state is well predicted by Romea’s theory. The equilibration of an unstable baroclinic jet is investigated in chapter 5. A predictive theory due to Esler based on global constraints is extended to test two new hypotheses, which are also shown to be successful in predicting the equilibrated flow profile of initially symmetric jets. The theory is adapted to include asymmetric initial jets where each hypothesis is found to have limited quantitative success.

This thesis deals with the following type of problems, which we denote partition problems, Given a set X in R^d, is there a way to partition X such that the convex hulls of all parts satisfy certain combinatorial properties? We focus on the following two kinds of partition problems. Tverberg type partitions. In this setting, one of the properties we ask the sets to satisfy is that their convex hulls all intersect. Ham sandwich type partitions. In this setting, one of the properties we ask the sets to satisfy is that the interior of their convex hulls are pairwise disjoint. The names for these types of partitions come from the quintessential theorem from each type, namely Tverberg's theorem and the ham sandwich theorem. We present a generalisation and a variation of each of these classic results. The generalisation of the ham sandwich theorem extends the classic result to partitions into any arbitrary number of parts. This is presented in chapter 2. Then, in chapter 3, variations of the ham sandwich theorem are studied when we search for partitions such that every hyperplane avoids an arbitrary number of sections. The generalisation of Tverberg's theorem consists of adding a condition of tolerance to the partition. Namely, that we may remove an arbitrary number of points and the partition still is Tverberg type. This is presented in chapter 4. Then, in chapter 5, ``colourful'' variations of Tverberg's theorem are studied along their applications to some purely combinatorial problems.

This thesis examines free surface flows in electrohydrodynamics under forcing in the form of a moving pressure distribution or topography. The ideas from examining free surface flows with forcing and those ideas andmethods coming from examining solitary waves within electrohydrodynamics are combined to study free surface flows under forcing in electrohydrodynamics. Chapter 1 gives a brief introduction to the ideas and work that have gone into investigating free surface flows and solitary waves in general and gives an idea of what will happen in the thesis. Chapter 2 formulates the general problem for the full nonlinear case and then examines the linear solution for both a moving pressure distribution and topography and presents profiles of the free surfaces and then shows that the solutions are nonuniform by examining the deep water case. Chapter 3 introduces the scaling for the weakly nonlinear problem and produces an equation which there is no nonuniformity and the amplitude of the free surface is finite. The case when the Bond number is around a 1/3 is also examined. Stokes analysis is performed to look for Wilton ripples. Chapter 4 examines conducting fluids adhering to an upper surface, the basic equations are set up and then the dispersion relation is derived to examine the existence of linear waves for certain values of the wavenumber k. A set of weakly nonlinear equations are examined and then solved numerically with examples of periodic profiles presented. A Stokes analysis is carried out for small amplitudes to look for Wilton ripples. An analysis is carried out for the approximation of long wavelength but finite depth, where the wave amplitude is the depth of the fluid. Chapter 5 considers surface flows and generalised the results from chapters 2 and 3 from two dimensions to three, linear free surface profiles are calculated and plotted and the weakly nonlinear equation is derived for the cases where the Bond number is close to 1/3 and not close to 1/3 giving a 5th order (Kadomstev-Petviashvili) (KP) equation. Chapter 6 is the set of conclusions and avenues of future research.

This thesis deals with the theory of integrable lattices in "solitons" throughout. Chapter 1 is a general introduction, which includes an historical survey and a short surrunary of the "solitons" theory and the present work. In Chapter 2, we discuss the equivalence between two kinds of lattice AKNS spectral problems - one includes two potentials, while the other includes four. The two nonlinear lattice systems associated with those two spectral problems, respectively is also proved to be equivalent to each other. In Chapter 3, we derive a class of nonlinear differential-difference equations (NDDEs) and put them into the Hamiltonian systems. Their complete integrability are proved in terms of so called "r-matrix". In the end of this Chapter, we study the symmetry properties and the related topics for lattice systems. In particular, we give detail for the Toda lattice systems. Chapter 4 is concerned with the Backlund transformations (BTs) and nonlinear superposition formulae (NSFs) for a class of NDDEs. A new method is presented to derive the generalized BTs and to prove that these BTs are precisely and really the auto-BTs. The three kinds of NSFs are derived by analysis of so called "elementary BTs". In Chapter 5, we investigate some relations between our lattices and the well-studied continuous systems. The continuum limits of our lattice systems and the discretizations of the continuous systems are discussed. The other study is about how we can consider a BT of continuous systems as a NDDE and then how a BT of such a NDDE can be reduced to the three kinds of NSFs of the continuous systems. The last Chapter is a study of integrable lattices under periodic boundary conditions. It provides a mathematical foundation for the study of integrable models in statistical mechanics. We are particularly interested in the lattice sine-Gordon and sinh-Gordon models. We not only prove the integrability of these models but also derive all kinds of classical phase shifts and some other physically interesting relations.

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