Höchste Magnetfelder haben sich zu einem unverzichtbaren Werkzeug der Festkörperphysik entwickelt. Sie werden insbesondere verwendet, um die elektronischen Eigenschaften von modernen Materialien zu erforschen. Da Magnetfelder oberhalb von 45 Tesla nicht mehr mit statischen (z.B. supraleitenden) Feldern zu erreichen sind, haben sich weltweit verschiedene Labore auf die Erzeugung gepulster Magnetfelder mit angestrebten Maximalwerten von 100 Tesla spezialisiert. In der vorliegenden Arbeit werden Anwendungsmöglichkeiten der kernmagnetischen Resonanz (NMR) in gepulsten Magnetfeldern aufgezeigt. Es ist gelungen, die starke Zeitabhängigkeit der gepulsten Magnetfelder mittels NMR präzise zu vermessen. Die genaue Kenntnis des Magnetfelds nach dem Puls ermöglicht, die Zeitabhängigkeit aus den Daten zu entfernen, sodass auch eine kohärente Signal-Mittelung möglich ist. Davon ausgehend werden erstmalig Messungen der chemischen Verschiebung, der Knight Shift, der Spin-Gitter-Relaxationsrate 1/T1 und der Spin-Spin-Relaxationsrate 1/T2 diskutiert. Schließlich werden die im Zusammenhang mit gepulsten Magnetfeldern erarbeiteten Gleichungen in vereinfachter Form zur genauen Messung und Analyse des freien Induktions-Zerfalls von 19F Kernspins in Calciumfluorid verwendet. Durch Messung des Zerfalls über sechs Größenordnungen wird eine genaue Analyse bezüglich einer neuartigen Theorie ermöglicht, welche den Zerfall basierend auf der Annahme mikroskopischen Chaos\'' erklärt. Diese Theorie hat das Potenzial, zu einem tieferen Verständnis von Quantenchaos in makroskopischen Vielteilchensystemen zu führen.

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Chaos theory holds that systems act in unpredictable nonlinear ways and that their behavior can only be observed, never predicted. This is an informative model for an organization in crisis. The West Virginia water contamination crisis, which began on January 9, 2014, fits the criteria of a system in chaos. Given the lack of appropriate response from the established organizations involved, many emergent organizations formed to help fill unmet informational and physical needs of the affected population. Crisis researchers have observed these ephemeral organizations for decades, but the recent proliferation of information communication technologies (ICT’s) have made their activities more widespread and observable. In West Virginia, their activities were indispensable to the affected population and helped restore a sense of normalcy. In this chaotic system, the emergent organizations functioned as strange attractors, helping move the system away from bifurcation and towards normalcy. This dissertation uses a qualitative approach to study the emergent organizations and how their presence and efforts were the mechanism that spurred the self-organization process.

This thesis deals with the application of optimisation based Validation and Verification (V&V) analysis on aerospace vehicles in order to determine their worst case performance metrics. To this end, three aerospace models relating to satellite and launcher vehicles provided by European Space Agency (ESA) on various projects are utilised. As a means to quicken the process of optimisation based V&V analysis, surrogate models are developed using polynomial chaos method. Surro- gate models provide a quick way to ascertain the worst case directions as computation time required for evaluating them is very small. A sin- gle evaluation of a surrogate model takes less than a second. Another contribution of this thesis is the evaluation of operational safety margin metric with the help of surrogate models. Operational safety margin is a metric defined in the uncertain parameter space and is related to the distance between the nominal parameter value and the first instance of performance criteria violation. This metric can help to gauge the robustness of the controller but requires the evaluation of the model in the constraint function and hence could be computationally intensive. As surrogate models are computationally very cheap, they are utilised to rapidly compute the operational safety margin metric. But this metric focuses only on finding a safe region around the nominal parameter value and the possibility of other disjoint safe regions are not explored. In order to find other safe or failure regions in the param- eter space, the method of Bernstein expansion method is utilised on surrogate polynomial models to help characterise the uncertain param- eter space into safe and failure regions. Furthermore, Binomial failure analysis is used to assign failure probabilities to failure regions which might help the designer to determine if a re-design of the controller is required or not. The methodologies of optimisation based V&V, surrogate modelling, operational safety margin, Bernstein expansion method and risk assessment have been combined together to form the WCAT-II MATLAB toolbox.

This thesis examines the intersection of math and art by focusing on three specific branches of math: the fourth dimension, non-Euclidean geometry, and chaos and fractals. Different genres of art interact with each of these branches of math. The influence of the fourth dimension can easily be seen in Cubism and Russian Constructivism. Non-Euclidean geometry guided some of M.C. Escher’s work, and it inspired the Crochet Coral Reef project. Chaos and fractals can be found in art and architecture throughout history, but Vincent van Gogh and Jackson Pollock are notable examples of artists who used chaos in their work. Some artists incorporate math into their work in a rigorous, exacting manner, while others take inspiration from a general concept and provide a more abstract interpretation. Regardless of mathematical accuracy, mathematically inspired art can provide a greater understanding of mathematical concepts.

In this work methods for performing time series prediction on complex real world time series are examined. In particular series exhibiting non-linear or chaotic behaviour are selected for analysis. A range of methodologies based on Takens' embedding theorem are considered and compared with more conventional methods. A novel combination of methods for determining the optimal embedding parameters are employed and tried out with multivariate financial time series data and with a complex series derived from an experiment in biotechnology. The results show that this combination of techniques provide accurate results while improving dramatically the time required to produce predictions and analyses, and eliminating a range of parameters that had hitherto been fixed empirically. The architecture and methodology of the prediction software developed is described along with design decisions and their justification. Sensitivity analyses are employed to justify the use of this combination of methods, and comparisons are made with more conventional predictive techniques and trivial predictors showing the superiority of the results generated by the work detailed in this thesis.

Computing reliable numerical approximations of invariant sets for nonlinear systems is the core problem for computer assisted study of dynamical systems. In the case of conservative systems the problem is complicated by the fact that there is no phase space dissipation to drive orbits onto attractors. In this dissertation we discuss several contributions to the field of computer assisted study of invariant dynamics in conservative systems. / text

Controlling the scrape-off layer (SOL) properties in order to limit divertor erosion and extend component lifetime will be crucial to successful operation of ITER and devices that follow, where intermittent thermal loads on the order of GW/m² are expected. Steady state transport in the edge region is generally turbulent with large, order unity, fluctuations and is convection dominated. Owing to the success of the past fifty years of progress in magnetically confining hot plasmas, in this work we examine convective transport phenomena in the SOL that occur in the relatively "slow", drift-ordered fluid limit, most applicable to plasmas near MHD equilibrium. Diamagnetic charge separation in an inhomogeneous magnetic field is the principal energy transfer mechanism powering turbulence and convective transport examined in this work. Two possibilities are explored for controlling SOL conditions. In chapter 2 we review basic physics underlying the equations used to model interchange turbulence in the SOL and use a subset of equations that includes electron temperature and externally applied potential bias to examine the possibility of suppressing interchange driven turbulence in the Texas Helimak. Simulated scans in E₀×B₀ flow shear, driven by changes in the potential bias on the endplates appears to alter turbulence levels as measured by the mean amplitude of fluctuations. In broad agreement with experiment negative biasing generally decreases the fluctuation amplitude. Interaction between flow shear and interchange instability appears to be important, with the interchange rate forming a natural pivot point for observed shear rates. In chapter 3 we examine the possibility of resonant magnetic perturbations (RMPs) or more generally magnetic field-line chaos to decrease the maximum particle flux incident on the divertor. Naturally occurring error fields as well as RMPs applied for stability control are known to cause magnetic field-line chaos in the SOL region of tokamaks. In chapter 3 2D simulations are used to investigate the effect of the field-line chaos on the SOL and in particular on its width and peak particle flux. The chaos enters the SOL dynamics through the connection length, which is evaluated using a Poincaré map. The variation of experimentally relevant quantities, such as the SOL gradient length scale and the intermittency of the particle flux in the SOL, is described as a function of the strength of the magnetic perturbation. It is found that the effect of the chaos is to broaden the profile of the sheath-loss coefficient, which is proportional to the inverse connection length. That is, the SOL transport in a chaotic field is equivalent to that in a model where the sheathloss coefficient is replaced by its average over the unperturbed flux surfaces. Both fully chaotic and the flux-surface averaged approximation of RMP application significantly lower maximum parallel particle flux incident on the divertor. / text

La tribologie et l'analyse non-linéaire du signal est le sujet de mon travail de thèse. Dans la nature, les phénomènes linéaires sont l'exception ; rares sont les systèmes réels qui obéissent exclusivement à des lois linéaires. A l'opposée, les non-linéarités sont impliquées dans tous les processus naturels (réactions chimiques, mécanique, économie, etc.). Les systèmes frottant en sont un des exemples les plus courants, avec des applications très variées. Dans les systèmes de freinage, le frottement se trouve être à l'origine de nombreux problèmes d'instabilités. Les types d'instabilités auxquelles on a affaire dans ce cas sont celles des vibrations induites par le frottement. C'est justement sur ces instabilités que repose le principe Modalsens; un capteur- une lamelle- vient frotter sur un échantillon et ce frottement génère la vibration de celui-ci : le post-traitement par analyse de Fourier du signal vibratoire permet de distinguer des composantes liées au relief, au frottement et à la compressibilité des aspérités. Dans le cas de la méthode Modalsens, l'analyse de Fourier, qui est un outil linéaire, agit comme des lunettes aux travers desquelles est observé le signal et qui filtrerait toutes les composantes non-linéaires. Notre contribution s'inscrit dans cette optique: mettre en place une méthode performante d'analyse non-linéaire pour permettre de mieux appréhender l'analyse du comportement dynamique de Modalsens et de dégager de nouveaux estimateurs pour la caractérisation des surfaces textiles. Partant de là, les résultats obtenus nous serviront à proposer une modélisation du contact sur matériaux fibreux.

Systems whose time evolutions are entirely deterministic can nevertheless be studied probabilistically, <em>i. e. </em> in terms of the evolution of probability distributions rather than individual trajectories. This approach is central to the dynamics of ensembles (statistical mechanics) and systems with uncertainty in the initial conditions. It is also the basis of ergodic theory--the study of probabilistic invariants of dynamical systems--which provides one framework for understanding chaotic systems whose time evolutions are erratic and for practical purposes unpredictable. Delay differential equations (DDEs) are a particular class of deterministic systems, distinguished by an explicit dependence of the dynamics on past states. DDEs arise in diverse applications including mathematics, biology and economics. A probabilistic approach to DDEs is lacking. The main problems we consider in developing such an approach are (1) to characterize the evolution of probability distributions for DDEs, <em>i. e. </em> develop an analog of the Perron-Frobenius operator; (2) to characterize invariant probability distributions for DDEs; and (3) to develop a framework for the application of ergodic theory to delay equations, with a view to a probabilistic understanding of DDEs whose time evolutions are chaotic. We develop a variety of approaches to each of these problems, employing both analytical and numerical methods. In transient chaos, a system evolves erratically during a transient period that is followed by asymptotically regular behavior. Transient chaos in delay equations has not been reported or investigated before. We find numerical evidence of transient chaos (fractal basins of attraction and long chaotic transients) in some DDEs, including the Mackey-Glass equation. Transient chaos in DDEs can be analyzed numerically using a modification of the "stagger-and-step" algorithm applied to a discretized version of the DDE.