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In this thesis the nonlinear and nonequilibrium properties of graphene are experimentally investigated using degenerate four--wave mixing and time--resolved pump--probe spectroscopy. High quality exfoliated natural graphite and large area epitaxial graphene on silicon carbide are investigated with femtosecond and picosecond ultrafast pulses in the near--infrared. A bespoke technique for suspending exfoliated graphene is also presented. In Chapter 3, the third--order nonlinear susceptibility of graphene is measured for the first time and shows a remarkably large response. Degenerate four--wave mixing at near--infrared wavelengths demonstrates an almost dispersionless emission over a broad spectral range. Quantum kinetic theory is employed to estimate the magnitude of the response and is in good agreement with the experimental data. The large susceptibility enables high contrast imaging, with a monolayer flake contrast of the order 10^{7} times higher than for standard reflection imaging. The degenerate four--wave mixing technique is utilised in Chapter 4 to measure the interface carbon signal of epitaxially grown graphene on silicon carbide. Comparable third--order signal from the silicon carbide bulk prevents true interface imaging. Excluding the third--order emission from detection by elongating the emission to outside a band--pass filter range allows for pure interfacial luminescence imaging. Features within the two growth faces are investigated with Raman spectroscopy. Nonlinear measurements are an increasingly popular tool for investigating fundamental properties of graphene. Chapter 5 investigates the influence of ultrafast pulses on the nonlinear response of graphene. High instantaneous intensities at the sample are shown to reduce the nonlinear emission by a factor or two. Comparing the Raman peak positions, widths and intensities before and after irradiation points to a huge doping of the samples, of the order 500 meV. In Chapter 6 the relaxation of photoexcited carriers is measured via time--resolved pump--probe spectroscopy, where a layer dependence of hot phonon decay is observed. Single layer flakes are observed to relax faster than bilayers and trilayers, with an asymptote reached at approximately four layers. Removing the substrate and measuring fully suspended samples reveals the same trend, suggesting that substrate interactions are not the cause of the enhanced decay. The decay mechanism is therefore intrinsic to graphene, perhaps due to coupling to out--of--plane, flexural phonons. The thickness dependence of epitaxial graphene on silicon carbide is compared to that of exfoliated flakes where the layer dependence is not observed. Phonon relaxation times, however, are in good agreement. Predictions for future investigations into this novel material based on the works here are suggested in Chapter 7. Preliminary pump--probe measurements at high carrier concentrations are an example of such progress, which will offer an insight into further decay mechanisms in graphene.

This thesis is concerned with long-range sound propagation in deep water.  The main area of interest is the stability of acoustic ray paths in wave guides in which there is a transition from single to double duct sound speed profiles, or vice-versa.  Sound propagation is modelled within a ray theoretical framework, which facilitates a dynamical systems approach of understanding long-range propagation phenomena, and the use of its tools of analysis. Alternative reduction techniques to the Poincaré sections are presented, by which the stability of acoustic rays can be graphically determined.  Beyond periodic driving, these techniques prove to be useful in case of the simplest quasiperiodic driving of the ray equations.  One of the techniques facilitates a special representation of ray trajectories for periodic driving. Namely, the space of sectioned trajectories is partitioned into nonintersecting regular and chaotic regions as with the Poincaré sections, when quasiperiodic and chaotic trajectories are represented by curve segments and area filling points, respectively.  In case of the simplest quasiperiodic driving – speaking about the same technique – regular trajectories are represented by curves similar to Lissajous curves, which are opened or closed depending on whether the two driving frequencies involved make relative primes or not. It is confirmed for a perturbed canonical profile that the background sound speed structure controls ray stability. It is also demonstrated for a particular double duct profile, when the singularity of the nonlinearity parameter for the homoclinic trajectory associated with this profile refers to the strong instability of corresponding perturbed trajectories.

The drill-strings used in drilling operate under extreme condi-tions, therefore, an accurate understanding of their dynamics is necessary and has attracted much interest. Although a bottom hole assembly (BHA) is to a great ex- tent responsible for the dynamics of the system, the in uence of the drill-pipes has been increasingly neglected by current models. Their dynamics and geometrical behaviour should be better analysed for a deeper understanding of underlying phe- nomena. For example, under stick-slip oscillations, the torque on the drill-string may cause torsional buckling of the drill-pipes, incurring in helical con guration, in which the apparent length is reduced, a ecting the forces at the bit{rock interface. With such behaviour and interactions in mind, this work focuses on elaborating a comprehensive mathematical model to investigate the dynamics of drill-strings, with attention to the drill-pipes section. Firstly, lower dimensional models are used to analyse the stick-slip limit cycle and its limits of existence. Then, a model developed for MEMS is used as a base for a comprehensive model using the formu- lation of Cosserat rods. Relevant boundary conditions are applied and a numerical simulation procedure is established. Simulations are performed for a range of sce- narios under stick-slip occurrence, and the behaviour of the drill-pipes is analysed. Focus is then given to axial vibrations with bit-bounce and the in uence on stick- slip, later to lateral vibrations with whirling motion of the drill-pipes, and nally to helical con gurations, taken by the drill-string under combined torsional, axial and lateral loads, showing the consequent shortening of the drill-string.

In this dissertation we study the anisotropic Manev problem that describes the motion of two point masses in an anisotropic space under the influence of a Newtonian force-law with a relativistic correction term. The dynamic of the system under discussion is very complicated and we use various methods to find a qualitative description of the flow. One of the strategies we use is to study the collision and near collision orbits. In order to do that we utilize McGehee type transformations that lead to an equivalent analytic system with an analytic energy relation. In these new coordinates the collisions are replaced by an analytic two-manifold: the so called collision manifold. We focus our attention on the heteroclinic orbits connecting fixed points on the collision manifold and on the homoclinic orbit to the equator of the mentioned manifold. We prove that as the anisotropy is introduced only four heteroclinic orbits persist and we show the exixtence of infinitely many transversal homoclinic orbits using a suitable generalization of the Poincaré-Melnikov method. Another strategy we apply is to study the symmetric periodic orbits of the system. To tackle this problem we follow two different approaches. First we apply the Poincaré continuation method and we find symmetric periodic orbits for small values of the anisotropy. Then we utilize a direct method of the calculus of variations, namely the lower semicontinuity method, and we prove the existence of symmetric periodic orbits for any value of the anisotropy parameter. In the last chapter we use the Killing's equation in an unusual way to prove that the anisotropic Kepler problem (that can be considered a particular case of the Manev) does not have first integrals linear in the momentum. / Graduate