An increasing number of contemporary studies in aquatic ecology emphasize the im- portance of highly unsaturated fatty acids (HUFAs) at the plant-animal interface. Studies have demonstrated a wide range of fatty acid profiles in primary producers, forcing her- bivorous zooplankton to differentially retain fatty acids to meet somatic requirements. Herbivores also vary in their somatic fatty acid profiles; cladocerans collect Eicosapen- taenoic Acid (EPA), copepods prefer Docosahexaenoic Acid (DHA). Fatty acid internal reserves can be broken down to meet structural needs (i.e. phospholipid synthesis), fuel reproduction and may play a role in cold weather adaptation. Several authors have noted increases in HUFA concentration with lowering ambient temperatures. Cladoceran membranes form a gel at lower temperatures, while copepod membranes remain fluid and allow active overwintering. Both fish and crustaceans accumulate high concentrations of HUFAs during periods of rapid growth, but colimitation with elemental resources may exist. Recent modeling results suggest food webs with high quality (nutritional and biochemical) primary producers can attain inverted biomass distributions with efficient energy transfer between trophic levels. The adoption rate for this material into man- agement studies remains low, and while other sectors of the scientific community thrive on the potential of HUFAs, planktonic food-web studies are choosing traditional view points over forward thinking. Bearing in mind the emerging hypotheses on the critical factors that drive the energy flow in the plant-animal interface, my dissertation will at- tempt to address the following general questions: What are the distinct signatures of food quality and food quantity on planktonic food web dynamics? How do nutritional and biochemical factors affect the flow of energy at the plant-animal interface? What is our current understanding of the role of highly unsaturated fatty acids (HUFAs) in aquatic food webs? To what extent can the current generation of plankton models reproduce the lower food web patterns when explicitly accounting for HUFAs? Is the integration of the HUFA role into water quality management models feasible? Explicitly accounting for HUFAs requires integrating factors of animal physiology with macro-ecology: what are the ramifications? Finally, what are the evolutionary aspects of animals coping with food quality?

An increasing number of contemporary studies in aquatic ecology emphasize the im- portance of highly unsaturated fatty acids (HUFAs) at the plant-animal interface. Studies have demonstrated a wide range of fatty acid profiles in primary producers, forcing her- bivorous zooplankton to differentially retain fatty acids to meet somatic requirements. Herbivores also vary in their somatic fatty acid profiles; cladocerans collect Eicosapen- taenoic Acid (EPA), copepods prefer Docosahexaenoic Acid (DHA). Fatty acid internal reserves can be broken down to meet structural needs (i.e. phospholipid synthesis), fuel reproduction and may play a role in cold weather adaptation. Several authors have noted increases in HUFA concentration with lowering ambient temperatures. Cladoceran membranes form a gel at lower temperatures, while copepod membranes remain fluid and allow active overwintering. Both fish and crustaceans accumulate high concentrations of HUFAs during periods of rapid growth, but colimitation with elemental resources may exist. Recent modeling results suggest food webs with high quality (nutritional and biochemical) primary producers can attain inverted biomass distributions with efficient energy transfer between trophic levels. The adoption rate for this material into man- agement studies remains low, and while other sectors of the scientific community thrive on the potential of HUFAs, planktonic food-web studies are choosing traditional view points over forward thinking. Bearing in mind the emerging hypotheses on the critical factors that drive the energy flow in the plant-animal interface, my dissertation will at- tempt to address the following general questions: What are the distinct signatures of food quality and food quantity on planktonic food web dynamics? How do nutritional and biochemical factors affect the flow of energy at the plant-animal interface? What is our current understanding of the role of highly unsaturated fatty acids (HUFAs) in aquatic food webs? To what extent can the current generation of plankton models reproduce the lower food web patterns when explicitly accounting for HUFAs? Is the integration of the HUFA role into water quality management models feasible? Explicitly accounting for HUFAs requires integrating factors of animal physiology with macro-ecology: what are the ramifications? Finally, what are the evolutionary aspects of animals coping with food quality?

We investigate the linear stability and perform the bifurcation analysis for Hopfield neural networks with a general distribution of delays, where the neurons are identical. We start by analyzing the scalar model and show what kind of information can be gained with only minimal information about the exact distribution of delays. We determine a mean delay and distribution independent stability region. We then illustrate a way of improving on this conservative result by approximating the true region of stability when the actual distribution is not known, but some moments or cumulants of the distribution are. We compare the approximate stability regions with the stability regions in the case of the uniform and gamma distributions. We show that, in general, the approximations improve as more moments or cumulants are used, and that the approximations using cumulants give better results than the ones using moments. Further, we extend these results to a network of n identical neurons, where we examine the stability of a symmetrical equilibrium point via the analysis of the characteristic equation both when the connection matrix is symmetric and when it is not. Finally, for the scalar model, we show under what conditions a Hopf bifurcation occurs and we use the centre manifold technique to determine the criticality of the bifurcation. When the kernel represents the gamma distribution with p=1 and p=2, we transform the delay differential equation into a system of ordinary differential equations and we compare the centre manifold computation to the one we obtain in the ordinary differential case.

Das Phänomen der Rissmusterentstehung stellt ein anschauliches und numerisch nachvollziehbares Beispiel für Strukturbildung dar. Im Rahmen der vorliegenden Arbeit gelang es, einfache dreidimensionale Rissmuster zu berechnen, die durch Zugspannungen infolge inhomogener Schrumpfung durch Wärme- oder Stofftransport wachsen. Aufgrund wechselseitiger Entlastung bleiben Risse oder Teile einer Rissfront zurück. Für idealisierte periodische Rissanordnungen wurden durch bruchmechanische Verzweigungsanalyse kritische Last- und Geometrieparameter berechnet, für die sich neben der Grundlösung eine weitere Lösung finden lässt. Dadurch konnte die Frage beantwortet werden, warum verschiedene Typen von Rissmustern entstehen. Untersucht wurden Rissmuster in trocknender Stärke und Basalt sowie Tunnelrisse zwischen Glasplatten. Dafür war es nötig, die Risskontur numerisch zu bestimmen, wofür ein neues iteratives Verfahren entwickelt wurde. Für Säulen bildende Rissmuster in Basalt und trocknender Stärke mit konstanten Säulendurchmessern, welche stationär getrieben wachsen, wurde gezeigt, dass entgegen der Annahme in der Literatur eine Verzweigungslösung für zusammenhängende dreidimensionale Rissmuster existiert. Die berechneten kritischen Rissabstände unterscheiden sich nur unwesentlich von denjenigen zweidimensionaler Rissmodelle. Zum Verständnis der Herausbildung der Basaltsäulen wurde ein neuer Mechanismus gefunden, für den die Wechselwirkung zwischen Riss- und Erstarrungsfront der flüssigen Lava eine wesentliche Rolle spielt. Mittels eines iterativen Verfahrens wurde dafür die Erstarrungsfront berechnet. Die Ergebnisse zeigen gute Übereinstimmung mit den experimentellen Werten. Die Durchmesser von Stärkesäulen und Basaltsäulen nehmen nach dem gleichen Potenzgesetz mit zunehmender Wachstumsgeschwindigkeit ab. Für Tunnelrisse sind die Rissabstände nahezu konstant, obwohl entgegen ursprünglicher Annahmen gezeigt werden konnte, dass diese instationär getrieben wachsen. Als Ursache dafür wurde die durch die Rissgeometrie stark reduzierte Wechselwirkung zwischen den Tunnelrissen erkannt. Der Rissabstand nimmt potenzartig mit Dicke der trocknenden Schicht zu. Mittels einer Skalenanalyse konnte gute Übereinstimmung mit den Experimenten gefunden werden. / Crack patterns are a remarkable example of pattern formation. In this work simple three-dimensional crack patterns where analysed that are driven by inhomogeneous shrinkage caused from heat or mass transport. Caused by mutual unloading, parts of the crack front stop to grow. For idealised periodically repeatable crack configurations through fracture mechanics bifurcation analysis critical loading or geometry parameters where calculated for which beside the basic solution (growth of all cracks) another solution is possible. Analysed were crack patterns in drying starch and basalt and tunnelling cracks between glass plates. For this it was necessary to calculate the crack contour numerically, for this a new iteration method was developed. For the connected crack patterns in starch and basalt that form columns, driven stationary, it could be shown that there exists a bifurcation solution. Agreement with experimental data was found by including the solidification front ahead the crack tips. For the tunnelling cracks the crack spacing is nearly constant also they are unsteadily driven. The crack spacing scales with a power law of the thickness of the drying layer. With a scaling analysis good agreement with experimental data was found.

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Many seemingly simple systems can display extraordinarily complex dynamics which has been studied and uncovered through nonlinear dynamical theory. The leitmotif of this thesis is changing phase-space structures and their (linear or nonlinear) stabilities by adding control functions (which act on the system as external perturbations) to the relevant Hamiltonians. These phase-space structures may be periodic orbits, invariant tori or their stable and unstable manifolds. One-electron systems and diatomic molecules are fundamental and important staging ground for new discoveries in nonlinear dynamics. In past years, increasing emphasis and effort has been put on the control or manipulation of these systems. Recent developments of nonlinear dynamical tools can provide efficient ways of doing so. In the first subtopic of the thesis, we are adding a control function to restore tori at prescribed locations in phase space. In the remainder of the thesis, a control function with parameters is used to change the linear stability of the periodic orbits which govern the processes in question. In this thesis, we report our theoretical analyses on multiphoton ionization of Rydberg atoms exposed to strong microwave fields and the dissociation of diatomic molecules exposed to bichromatic lasers using nonlinear dynamical tools. This thesis is composed of three subtopics. In the first subtopic, we employ local control theory to reduce the stochastic ionization of hydrogen atom in a strong microwave field by adding a relatively small control term to the original Hamiltonian. In the second subtopic, we perform periodic orbit analysis to investigate multiphoton ionization driven by a bichromatic microwave field. Our results show quantitative and qualitative agreement with previous studies, and hence identify the mechanism through which short periodic orbits organize the dynamics in multiphoton ionization. In addition, we achieve substantial time savings with this approach. In the third subtopic we extend our periodic orbit analysis to the dissociation of diatomic molecules driven by a bichromatic laser. In this problem, our results based on periodic orbit analysis again show good agreement with previous work, and hence promise more potential applications of this approach in molecular physics.

Orientador: Laurence Duarte Colvara / Banca: Carlos Roberto Minussi / Banca: Wagner Peron Ferreira / Resumo: Este trabalho apresenta um estudo sobre a relação entre os principais tipos de bifurcações que ocorrem em sistemas elétricos de potência e em quais ocasiões elas podem ocorrer em máquinas síncronas com ou sem RAT (Regulador Automático de Tensão). Para explorar tais fenômenos, primeiramente o sistema é modelado, sendo utilizado para o caso MBI (Máquina - Barramento In nito) o modelo um eixo e, então, a matriz de estado é calculada para a análise dos autovalores. Para os sistemas multimáquinas estudados, são incluídos dois enrolamentos amortecedores nos eixos d ¡ q. São então apresentados os métodos de análise de estabilidade transitória convencionais, amplamente utilizados, conhecidos como método Tradicional e Método Direto. As condições para a ocorrência de bifurcações são analisadas utilizando os coe - cientes linearizados do modelo de He ron-Phillips para o caso MBI, onde é mostrado que se espera perder a estabilidade para o sistema com regulador automático de tensão através de uma bifurcação de Hopf e para o caso sem RAT através de uma bifurcação Sela-Nó. Por m, é analisado o ciclo-limite para o caso de uma máquina - barramento in nito e para sistemas multimáquinas através do modelo não-linear. A região de estabilidade é analisada no plano de fase, sendo mostrada a necessidade de incluir a variação de uxo no enrolamento de campo para uma análise correta da estabilidade. É também mostrado que este ciclo-limite pode reduzir a fronteira de estabilidade calculada pelo método convencional. / Abstract: The aim of this study is the relation among main types of bifurcations that occur in electrical power systems and the circumstances they can happen with the synchronous machines considered with or without AVR (Automatic Voltage Regulator). To explore such phenomena, the system is rst modeled with the synchronous machines described by the one axis model for the MIB (Machine - In nite Bus) case , and so the state matrix is computed for the analysis of its eigenvalues. For multimachine systems case two windings dampers are included in d-q axes. The conditions for the occurrence of bifurcations are analyzed using the coe cients of the He ron-Phillips model for MIB case, where it is shown that one expects the system with automatic voltage regulator lose synchronism through a Hopf bifurcation and for the case without RAT through a Saddle- Node Bifurcation. Finally, the nonlinear model is accounted for in order to consider the limit-cycle for the case of one machine - in nite bus case as well as for multimachine system. Since internal voltage a ects the boundary of the stability region it must be considered. Then the phase portrait does not su ce and the trajectories must to be observed in a sub space de ned with the internal voltage. It is also shown that this limit-cycle can reduce the boundary of stability calculated by means of the direct method. / Mestre

[pt] A análise das ressonâncias internas em sistemas estruturais contínuos é uma das principais áreas de pesquisa no campo da dinâmica não linear. A ressonância interna entre dois modos de vibração ocorre quando a proporção de suas frequências naturais é um número inteiro. De particular importância, devido à sua influência na resposta estrutural, é a ressonância interna 1:1, geralmente associada às simetrias do sistema, a ressonância interna 1:2, devida às não linearidades quadráticas e a ressonância 1:3 decorrente de não linearidades cúbicas. A ressonância interna permite a transferência de energia entre os modos de vibração relacionados, levando geralmente a novos fenômenos com profunda influência sobre a estabilidade da resposta dinâmica. As cascas de revolução geralmente exibem ressonâncias internas devido à inerente simetria circunferencial e um denso espectro de frequência em sua faixa de frequências mais baixas. Isso pode levar não apenas a ressonâncias internas do tipo m:n, mas a múltiplas ressonâncias internas. Nesta tese é realizada a análise de múltiplas ressonâncias internas em cascas cilíndricas delgadas, em particular as ressonâncias internas de 1:1:1:1 e 1:1:2:2 são investigadas em detalhes, um tópico pouco explorado na literatura técnica. A investigação de ressonâncias internas em sistemas contínuos geralmente é realizada usando modelos discretos de baixa dimensão. Embora alguns trabalhos anteriores tenham investigado ressonâncias internas do tipo m:n em cascas cilíndricas, muitos resultados não são consistentes, uma vez que os modelos discretos derivados não consideram os acoplamentos modais devido a não linearidades quadráticas e cúbicas. Aqui, usando um procedimento de perturbação, expansões modais consistentes são derivadas para um número arbitrário de modos de interação, levando a modelos de baixa dimensão confiáveis. A precisão desses modelos é corroborada usando o método Karhunen-Loève. Finalmente, é bem sabido que pequenas imperfeições geométricas da ordem da espessura da casca têm uma forte influência na sua resposta não linear. No entanto, sua influência nas ressonâncias internas, instabilidade dinâmica e transferência de energia é desconhecida. Assim, a influência de diferentes tipos de imperfeição modal é devidamente considerada na presente análise. Utilizando os modelos discretos aqui derivados, é apresentada uma análise detalhada das bifurcações, utilizando técnicas de continuação e o critério de estabilidade de Floquet, esclarecendo a importância das ressonâncias internas nas vibrações não lineares e instabilidades de cascas cilíndricas. Os resultados também confirmam que a forma e a magnitude das imperfeições geométricas iniciais têm uma influência profunda nos resultados, permitindo ou impedindo a transferência de energia entre os modos ressonantes considerados. / [en] The analysis of internal resonances in continuous structural systems is one of the main research areas in the field of nonlinear dynamics. Internal resonance between two vibration modes occur when the ratio of their natural frequencies in an integer number. Of particular importance, due to its influence on the structural response, is the 1:1 internal resonance, usually associated with system symmetries, the 1:2 internal resonance, due to quadratic nonlinearities, and the 1:3 resonance arising from cubic nonlinearities. The internal resonance enables the energy transfer between the related vibration modes, leading usually to new phenomena with profound influence on the stability of the dynamic response. Shells of revolution usually exhibit internal resonances due to the inherent circumferential symmetry and a dense frequency spectrum in their lower frequency range. This may lead not only to m:n internal resonances, but also multiple internal resonances. In this thesis, the analysis of multiple internal resonances in slender cylindrical shells is conducted, in particular 1:1:1:1 and 1:1:2:2 internal resonances are investigated in detail, a topic rarely found in the technical literature. The investigation of internal resonances in continuous systems is usually conducted using low dimensional discrete models. Although some previous works have investigated m:n internal resonances in cylindrical shells, many results are not consistent since the derived discrete models do not consider the modal couplings due to quadratic and cubic nonlinearities. Here, using a perturbation procedure, consistent modal expansions are derived for an arbitrary number of interacting modes, leading to reliable low dimensional models. The accuracy of these models is corroborated using the Karhunen-Loève method. Finally, it is well known that small geometric imperfections of the order of the shell thickness has a strong influence on the shell nonlinear response. However, their influence on internal resonances, dynamic instability and energy transfer is largely unknown. Thus, the influence of different types of modal imperfection is properly considered in the present analysis. Using the derived discrete models, a detail bifurcation analysis, using continuation techniques and Floquet stability criterion, is presented, clarifying the importance of internal resonances on the nonlinear vibrations and instabilities of cylindrical shells. The results also confirm that the form and magnitude of initial geometric imperfections has a profound influence on the results enabling or preventing the energy transfer among the considered resonant modes.

Les écoulements vrillés, qui sont largement utilisés dans les turbines à gaz, sont connus pour être sujet à des bifurcations entre différentes topologies (grandes reconfigurations de l'écoulement) qui peuvent affecter les performances et la sécurité du moteur. Ce travail se concentre sur l'étude de ces bifurcations en utilisant la Simulation aux Grandes Echelles (SGE). Cette étude montre qu'un petit changement dans les conditions dynamique du fluide, induite par les différents modèles de sous-maille utilisés, peut provoquer une transition entre deux régimes d'écoulement distincts lorsque l'écoulement tourbillonnaire est proche des conditions critiques de transition. La sensibilité de la SGE aux modèles de sous-maille est également identifiée comme le résultat d'un manque de résolution à certains endroits critiques, un problème qui est analysé en utilisant une méthode d'adaptation de maillage. L’adaptation de maillage est testée sur des cas académiques et industriels. Ici, par ajustement de la résolution du maillage sur la base des caractéristiques de l'écoulement étudié (raffinement et grossissement de la grille en maintenant constant le coût numérique), des améliorations substantielles peuvent être obtenues, en terme de prédictions de la SGE. Ce travail peut être considéré comme une des premières étapes vers la mise en place d'une procédure standard (reproductible et indépendante de l’utilisateur) de maillage pour la SGE. / Swirling flows, which are widely employed in gas turbines, are known to undergo bifurcation between different topologies (large reconfigurations of the flow field) affecting the engine performance and safety. This work focuses on the study of such bifurcations using Large-Eddy Simulation (LES). It shows that a small change in the fluid dynamics conditions, induced by the different Sub-Grid Scale (SGS) models used in the simulations, can cause a transition between two, distinct, flow states when the swirling flow is close to transition conditions. The sensitivity of LES to SGS modeling is also identified as the result of a lack of mesh resolution at some critical locations, a problem which is analyzed using mesh adaptation. Mesh adaptation is tested on canonical and industrial flows. Here, by adjusting the mesh resolution based on the characteristics of the flow examined (refining and coarsening the grid keeping constant the numerical cost), substantial improvements of the LES predictions can be obtained. This work can be considered as the first step toward the establishment of a standard (repeatable and user independent) meshing procedure for LES.