<p>Ocean-based wireless sensor networks serve many important purposes ranging from tsunami early warning to anti-submarine warfare. Developing energy harvesting devices that make these networks self-sufficient allows for reduced maintenance cost and greater reliability. Many methods exist for powering these devices, including internal batteries, photovoltaic cells and thermoelectric generators, but the most reliable method, if realized, would be to power these devices with an internal kinetic energy harvester capable of reliably converting wave motion into electrical power. Designing such a device is a challenge, as the ocean excitation environment is characterized by shifting frequencies across a relatively wide bandwidth. As such, traditional linear kinetic energy harvesting designs are not capable of reliably generating power. Instead, a nonlinear device is better suited to the job, and the task of this dissertation is to investigate the behaviors of devices that could be employed to this end.</p><p>This dissertation is motivated by the design and analysis of an ocean energy harvester based on a horizontal pendulum system. In the course of investigating the dynamics of this system, several discoveries related to other energy harvesting systems were made and are also reported herein. It is found that the most reliable method of characterizing the behaviors of a nonlinear energy harvesting device in the characteristically random forcing environment of the ocean is to utilize statistical methods to inform the design of a functional device. It is discovered that a horizontal pendulum-like device could serve as an energy harvesting mechanism in small self-</p><p>sufficient wireless sensor buoys if properly designed and if the proper transduction mechanisms are designed and employed to convert the mechanical energy of the device into electrical power.</p> / Dissertation

In this thesis, bifurcation phenomena in dynamical systems with cosymmetry and Hamiltonian structure were investigated using numerical methods. Several numerical continuation methods and test functions for detecting bifurcations were presented. The numerical results for various examples are given using a numerical bifurcation analysis toolbox.

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As equações diferencias parciais tem origem na modelagem do problemas nas ciências e engenharia, tais como a equação do calor, equação da onda, equação de Poisson, entre outras. Para muitas destas equações não é tão simples obter uma técnica analítica para achar sua solução e nestes casos é necessário uso de soluções aproximadas obtidas pelo computador. Existem técnicas tradicionais para solução numérica de uma grande classe de equações diferenciais, mas quando esta equação está na forma implícita, muitas destas técnicas já não podem ser aplicadas. Frequentemente as equações diferenciais parciais de segunda ordem tem maior estudo que as equações de primeira ordem sendo uma das razões que os modelos envolvem derivadas de segunda ordem. No caso das equações diferenciais parciais de primeira ordem implícitas a não linearidade em alguns casos não permite determinar uma solução de forma simples. O trabalho desenvolvido faz uma revisão do método das características para estabelecer as condições necessárias e suficientes, que permitam encontrar uma solução, ao mesmo tempo evidencia a complexidade de determinar uma solução clássica. Dentro das aplicações existentes relacionadas com as Equações Diferenciais Parciais Implícitas de Primeira Ordem, podemos mencionar a Equação cinemática e a Equação de Hamilton-Jacobi que podem-se associar com o movimento de partículas. Para a solução de uma Equação Diferencial Implícita de Primeira Ordem o método das características tem uma estrutura de solução que permite resolver a equação de forma analítica e numérica, desde que se verifique o Teorema de Cauchy. O objetivo deste trabalho de mestrado é obter um método numérico para a solução de equações diferenciais parciais de primeira ordem implícitas. Nós propomos um método numérico do tipo previsor-corretor que resolve uma EDP de primeira ordem implícita, utilizando o sistema característico em conjunto com as condições de banda, para reduzir o erro global nas iterações. / Partial differential equations arise in the modeling of problems in science and engineering, such as the heat equation, wave equation, Poisson equation, among others. For many of these equations it is not so simple to obtain an analytical technique to find a solution in these cases and it is necessary to use a computer to obtain approximate solutions. There are traditional techniques for numerical solution of a large class of differential equations, but when this equation is in implicit form, many of these techniques can no longer be applied. Often partial differential equations of second order are more studied than first order equations the reason being that one of the models involve secondorder derivatives. In the case of implicit partial differential equations of first order the non-linearity in some cases does not allow for a solution in simple from to be determined. The work reviews the method of characteristics to establish the necessary and sufficient conditions that will find a solution at the same time demonstrates the complexity of determining classical solution. Within existing applications related to Partial Differential Equations of First Order Implicit, we can mention the textit kinematic equation and textit equation Hamilton-Jacobi that can be associated with the movement of particles. For the solution of a differential equation First Implicit Order the method of characteristics has a solution framework that enables solve the equation analytically and numerically, provided there is the Cauchy theorem. The objective of this master thesis is to obtain a numerical method for the solution of partial differential equations first order implicit. We propose a numerical method of predictor-corrector type that resolves a EDP first implicate order, using the characteristic system in conjunction with the band conditions, to reduce the overall error in iterations.

As equações diferencias parciais tem origem na modelagem do problemas nas ciências e engenharia, tais como a equação do calor, equação da onda, equação de Poisson, entre outras. Para muitas destas equações não é tão simples obter uma técnica analítica para achar sua solução e nestes casos é necessário uso de soluções aproximadas obtidas pelo computador. Existem técnicas tradicionais para solução numérica de uma grande classe de equações diferenciais, mas quando esta equação está na forma implícita, muitas destas técnicas já não podem ser aplicadas. Frequentemente as equações diferenciais parciais de segunda ordem tem maior estudo que as equações de primeira ordem sendo uma das razões que os modelos envolvem derivadas de segunda ordem. No caso das equações diferenciais parciais de primeira ordem implícitas a não linearidade em alguns casos não permite determinar uma solução de forma simples. O trabalho desenvolvido faz uma revisão do método das características para estabelecer as condições necessárias e suficientes, que permitam encontrar uma solução, ao mesmo tempo evidencia a complexidade de determinar uma solução clássica. Dentro das aplicações existentes relacionadas com as Equações Diferenciais Parciais Implícitas de Primeira Ordem, podemos mencionar a Equação cinemática e a Equação de Hamilton-Jacobi que podem-se associar com o movimento de partículas. Para a solução de uma Equação Diferencial Implícita de Primeira Ordem o método das características tem uma estrutura de solução que permite resolver a equação de forma analítica e numérica, desde que se verifique o Teorema de Cauchy. O objetivo deste trabalho de mestrado é obter um método numérico para a solução de equações diferenciais parciais de primeira ordem implícitas. Nós propomos um método numérico do tipo previsor-corretor que resolve uma EDP de primeira ordem implícita, utilizando o sistema característico em conjunto com as condições de banda, para reduzir o erro global nas iterações. / Partial differential equations arise in the modeling of problems in science and engineering, such as the heat equation, wave equation, Poisson equation, among others. For many of these equations it is not so simple to obtain an analytical technique to find a solution in these cases and it is necessary to use a computer to obtain approximate solutions. There are traditional techniques for numerical solution of a large class of differential equations, but when this equation is in implicit form, many of these techniques can no longer be applied. Often partial differential equations of second order are more studied than first order equations the reason being that one of the models involve secondorder derivatives. In the case of implicit partial differential equations of first order the non-linearity in some cases does not allow for a solution in simple from to be determined. The work reviews the method of characteristics to establish the necessary and sufficient conditions that will find a solution at the same time demonstrates the complexity of determining classical solution. Within existing applications related to Partial Differential Equations of First Order Implicit, we can mention the textit kinematic equation and textit equation Hamilton-Jacobi that can be associated with the movement of particles. For the solution of a differential equation First Implicit Order the method of characteristics has a solution framework that enables solve the equation analytically and numerically, provided there is the Cauchy theorem. The objective of this master thesis is to obtain a numerical method for the solution of partial differential equations first order implicit. We propose a numerical method of predictor-corrector type that resolves a EDP first implicate order, using the characteristic system in conjunction with the band conditions, to reduce the overall error in iterations.

Polynomial continuation, a branch of numerical continuation, has been applied to several primary problems in kinematic geometry. The objective of the research presented in this document was to explore the possible extensions of the application of polynomial continuation, especially in the field of deployable structure design. The power of polynomial continuation as a design tool lies in its ability to find all solutions of a system of polynomial equations (even positive dimensional solution sets). A linkage design problem posed in polynomial form can be made to yield every possible feasible outcome, many of which may never otherwise have been found. Methods of polynomial continuation based design are illustrated here by way of various examples. In particular, the types of deployable structures which form planar rings, or frames, in their deployed configurations are used as design cases. Polynomial continuation is shown to be a powerful component of an equation-based design process. A polyhedral homotopy method, particularly suited to solving problems in kinematics, was synthesised from several researchers' published continuation techniques, and augmented with modern, freely available mathematical computing algorithms. Special adaptations were made in the areas of level-k subface identification, lifting value balancing, and path-following. Techniques of forming closure/compatibility equations by direct use of symmetry, or by use of transfer matrices to enforce loop closure, were developed as appropriate for each example. The geometry of a plane symmetric (rectangular) 6R foldable frame was examined and classified in terms of Denavit-Hartenberg Parameters. Its design parameters were then grouped into feasible and non-feasible regions, before continuation was used as a design tool; generating the design parameters required to build a foldable frame which meets certain configurational specifications. Two further deployable ring/frame classes were then used as design cases: (a) rings which form (planar) regular polygons when deployed, and (b) rings which are doubly plane symmetric and planar when deployed. The governing equations used in the continuation design process are based on symmetry compatibility and transfer matrices respectively. Finally, the 6, 7 and 8-link versions of N-loops were subjected to a witness set analysis, illustrating the way in which continuation can reveal the nature of the mobility of an unknown linkage. Key features of the results are that polynomial continuation was able to provide complete sets of feasible options to a number of practical design problems, and also to reveal the nature of the mobility of a real overconstrained linkage.

Les instabilités hydrodynamiques en cavité chauffée latéralement jouent un rôle important dans certains processus de fabrication de matériaux tels que le procédé de Bridgman horizontal. En effet, le fluide (métal liquide qui va se solidifier) est le siège d’une circulation thermoconvective due à l’existence d’un gradient de température horizontal qui est susceptible de devenir instationnaire via des instabilités oscillatoires. La connaissance et la maîtrise de ces instabilités sont donc primordiales afin de pouvoir améliorer la qualité des cristaux obtenus par cette technique. Dans cette thèse, nous nous sommes intéressés en premier aux instabilités affectant la circulation convective dans une cavité tridimensionnelle de dimensions 4×2×1. (longueur × largeur × hauteur). Grâce aux techniques numériques de continuation, nous avons pu obtenir les solutions stationnaires et oscillatoires, ainsi que leur stabilité, jusqu’à l’apparition de la quasi-périodicité en fonction du nombre de Grashof Gr et pour un nombre de Prandtl allant de 0 à 0,025. Ensuite, pour un éventuel contrôle des instabilités, nous nous sommes intéressés aux effets induits par la rotation de la cavité. Nous avons tout d’abord considéré un modèle monodimensionnel que nous avons développé durant cette thèse. Ce modèle analytique, bien que simplifié, est en très bon accord avec les observations en dynamique des écoulements atmosphériques (déviation des masses fluides vers la droite de la composante de vitesse dominante et vents thermiques). La stabilité linéaire de cet écoulement est ensuite effectuée en fonction du taux de rotation donné par le nombre de Taylor et du nombre de Grashof pour un nombre de Prandtl allant de 0 à 10. Nous avons pu montrer à travers ce modèle que la rotation possède un caractère stabilisant vis-à-vis de ce type d’écoulement. Enfin, nous nous sommes focalisés sur les effets de la rotation sur l’écoulement pleinement tridimensionnel dans la cavité de dimensions 4×2×1. Nous avons mis en évidence deux régimes d’écoulements : un régime dominé par la convection, où la circulation du fluide est déviée par la rotation dans la diagonale de la cavité, et un deuxième régime dominé par la rotation où la circulation du fluide est concentrée dans les couches limites dites d’Ekman et de Stewartson. Un très bon accord est observé entre le modèle analytique simplifié et la simulation numérique tridimensionnelle. / Hydrodynamic instabilities in laterally heated cavities play an important role in some material processing techniques such as the horizontal Bridgman process. Indeed, the fluid (liquid metal to be solidified) is the seat of a thermoconvective circulation due to the existence of a horizontal temperature gradient which is likely to become unsteady via oscillatory instabilities. The knowledge and the control of these instabilities are thus essential in order to be able to improve the quality of the crystals obtained by this technique. In this thesis, we are first interested in the instabilities of the convective circulation in a three-dimensional cavity of dimensions 4×2×1 (length × width × height). Thanks to the numerical continuation techniques, we were able to obtain the stationary and oscillatory solutions, as well as their stability, until the appearance of the quasi-periodicity according to the Grashof number Gr and for a Prandtl number Pr ranging from 0 to 0.025.Then, the effects induced by a rotation of the cavity around the vertical axis parallel to gravity (for a possible control of the instabilities) are studied and a one-dimensional model developed during this thesis was first considered. This analytical model, although simplified, is in very good agreement with the observations of the atmospheric flows (deviation of the fluid masses towards the right of the component of the dominant velocity and thermal winds). The linear stability of this flow as well as an energy analysis at the thresholds are then performed as a function of the rotation rate given by the Taylor number Ta and the Grashof number Gr for a Prandtl number Pr ranging from 0 to 10. Through this model, we have been able to show that the rotation has a stabilizing effect on this type of flow.We finally focused on the effects of this type of rotation on the steady fully threedimensional flow observed in the cavity 4×2×1 at low Grashof numbers.We have highlighted two flow regimes: a regime dominated by convection where the fluid circulation, deviated by the rotation, occurs in the diagonal of the cavity, and a second regime dominated by rotation where the fluid circulation is concentrated in the so-called Ekman and Stewartson boundary layers. A very good agreement is observed between the simplified analytical model and the three-dimensional numerical simulation.

We study fast-slow versions of the SIR, SIRS and SIRWS epidemiological models, and of the SIRS epidemiological model on homogeneous graphs, obtained through the application of the moment closure method. The multiple time scale behavior is introduced to account for large differences between some of the rates of the epidemiological pathways. Our main purpose is to show that the fast-slow models, even though in nonstandard form, can be studied by means of Geometric Singular Perturbation Theory (GSPT). In particular, without using Lyapunov's method, we are able to not only analyze the stability of the endemic equilibria of the SIR and SIRS models, but also to show that in the remaining models limit cycles arise. We show that the proposed approach is particularly useful in more complicated (higher dimensional) models such as the SIRWS model and the SIRS on homogeneous graphs, for which we provide a detailed description of their dynamics by combining analytic and numerical techniques. In particular, for the latter we show that the model can give rise to periodic solutions, differently from the corresponding model based on homogeneous mixing.

Cette thèse porte sur un problème couplé des lanceurs, à savoir une manœuvre de l'attitude couplée avec la trajectoire minimisant le temps de manœuvre. La difficulté de ce problème vient essentiellement du phénomène de chattering et du couplage des dynamiques n'ayant pas la même échelle de temps. Avec une analyse géométrique des extrémales venant de l'application du principe du maximum de pontryagin, nous donnons des conditions suffisantes sous lesquelles le phénomène de chattering se produit, pour des systèmes affines bi-entrée. Nons appliquons ensuite ce résultat à notre problème, et montrons que le phénomène de chattering arrive pour les trajectoires optimales, pour certaines données terminales. A l'aide de cette analyse théorique préliminaire, nous mettons en œuvre une méthode de résolution indirecte efficace, combinée à une méthode de continuation prédicteur-correcteur. En cas de chattering, deux stratégies sous-optimales sont proposées: soit une méthode directe dont le contrôle est approché par un contrôle constant par morceaux, soit en stoppant la continuation avant l'échec dû au chattering. Avec le tir multiple et plusieurs paramètres de continuations supplémentaires, cette méthode de résolution est appliquée à chercher une manœuvre de pull-up avec des contraintes sur l'état en minimisant le temps-énergie pour des lanceurs aéroportés. Les résultats numériques permettent de mettre en évidence l'efficacité et la robustesse de notre méthode de résolution. / In this thesis, we investigate the minimum time control problem for the control and guidance of a launch vehicle, whose motion is described by its attitude kinematics and dynamics but also by its trajectory dynamics. The difficulty of this problem is essentially due to the chattering phenomenon and to the coupling of dynamics of different time scales. With a refined geometric study of the extremals coming from the application of the pontryagin maximum principle, we establish a general result for bi-input control-affine systems, providing sufficient conditions under which the chattering phenomenon occurs. We show how this result can be applied to our problem. Based on this preliminary theoretical analysis, we implement an efficient indirect numerical method, combined with numerical predictor-corrector continuation, in order to compute numerically the optimal solutions of the problem. In case of chattering, two sub-optimal strategies are designed: one is a direct method in which the control is approximated by a piecewise constant control, and the other consists of stopping the continuation procedure before its failure due to chattering. With several additional numerical continuation steps, we apply finally the developed indirect approach to the minimum time-energy pull-up maneuver problem, in which state constraints are also considered, for airborne launchers. Numerical simulations illustrate the efficiency and robustness of our method.

La diversité des régimes des instruments de la famille des flûtes a été mise en évidence à de nombreuses reprises : régimes statiques, périodiques, ou non périodiques. Cependant, de nombreux aspects de la dynamique de ces instruments demeurent mal compris. Pour les musiciens comme pour les facteurs d'instruments, les transitions entre régimes revêtent une importance particulière : d'une part elles correspondent à des changements de notes, et d'autre part la production d'un régime donné est conditionnée par les paramètres de facture (liés à la fabrication de l'instrument), et de contrôle (ajustés en permanence par l'instrumentiste). On s'attache dans ce document à caractériser les transitions entre régimes dans les flûtes, en lien avec des problématiques de facture et de jeu. Différentes approches sont mises en place. Des approches expérimentales d'une part, avec des mesures sur musicien et sur bouche artificielle. Par ailleurs, un modèle physique de l'instrument - un système dynamique à retard de type neutre - est étudié, par intégration temporelle d'une part, mais également par collocation orthogonale et continuation, donnant ainsi accès aux diagrammes de bifurcations.Croiser les résultats de ces différentes approches permet de mieux appréhender différents phénomènes : hystérésis associée aux changements de régime, ou mécanisme d'apparition des régimes non périodiques. L'influence de paramètres de facture et de contrôle est également étudiée : le rôle majeur de la géométrie interne du canal des flûtes à bec est mis en évidence, et l'influence de la dynamique de la pression dans la bouche du musicien sur les seuils de changement de régimes est caractérisée. / Various studies have highlighted the diversity of regimes in flute-like instruments : static, periodic or non periodic regimes. However, some aspects of their dynamics remain poorly understood. Both for flute players and makers, transitions between regimes are particularly important : on the one hand, they correspond to a change of the note played, and on the other hand, production of a given regime is determined by parameters related to making and to playing of the instrument. In this document, we are interested in caracteristics of regime change in flute-like instruments, in relation with making and playing issues.Different approches are considered. First, experimental methods, with measurement on both musician and an artificial mouth. On the other hand, a physical model of the instrument - a system of delay differential equations of neutral type - is studied, through time-domain integration, and using orthogonal collocation coupled to numerical continuation. This last approach provides access to bifurcation diagrams.Considering results of these different methods, it becomes possible to better understand different experimental phenomena, such as regime change and associated hysteresis, or production mechanisms of non periodic regimes. Influence of different parameters is further studied : the crucial importance of the channel geometry in recorders is highlighted, and the influence of the mouth pressure dynamics on regime change thresholds is analysed.