Piecewise smooth maps appear as models of various physical, economical and other systems. In such maps bifurcations can occur when a fixed point or periodic orbit crosses or collides with the border between two regions of smooth behaviour as a system parameter is varied. These bifurcations have little analogue in standard bifurcation theory for smooth maps and are often more complex. They are now known as "border collision bifurcations". The classification of border collision bifurcations is only available for one-dimensional maps. For two and higher dimensional piecewise smooth maps the study of border collision bifurcations is far from complete. In this thesis we investigate some of the bifurcation phenomena in two-dimensional continuous piecewise smooth discrete-time systems. There are a lot of studies and observations already done for piecewise smooth maps where the determinant of the Jacobian of the system has modulus less than 1, but relatively few consider models which allow area expansions. We show that the dynamics of systems with determinant greater than 1 is not necessarily trivial. Although instability of the systems often gives less useful numerical results, we show that snap-back repellers can exist in such unstable systems for appropriate parameter values, which makes it possible to predict the existence of chaotic solutions. This chaos is unstable because of the area expansion near the repeller, but it is in fact possible that this chaos can be part of a strange attractor. We use the idea of Markov partitions and a generalization of the affine locally eventually onto property to show that chaotic attractors can exist and are fully two-dimensional regions, rather than the usual fractal attractors with dimension less than two. We also study some of the local and global bifurcations of these attracting sets and attractors.Some observations are made, and we show that these sets are destroyed in boundary crises and some conditions are given.Finally we give an application to a coupled map system.

In this dissertation, the mechanical models of 2-DOF and 4-DOF nonlinear journal bearing rotor systems are established. A more accurate model of oil film forces is derived from Reynolds equations. The periodic motions in such nonlinear journal bearing systems are obtained through discrete mapping method. Such a semi-analytical method constructs an implicit discrete mapping structure for periodic motions by discretization of the continuous journal bearing rotor differential equations. Stable and unstable periodic solutions of periodic motions are obtained with prescribed accuracy. The bifurcation tree of periodic motions in rotor system without oil film forces is demonstrated through the route from period-1 motion to period-8 motion. Stable period-2 and unstable period-1 motion are presented for 2 DOF journal bearing rotor system. Possibly infinite periodic solutions are found in 4 DOF journal bearing rotor system. For the rotor systems, the stability and bifurcations of periodic motions are analyzed through eigenvalue analysis of the corresponding Jacobian matrix of the discretized nonlinear systems. The frequency amplitude characteristics of periodic motions in 2 DOF journal bearing system are presented for a good understanding of the nonlinear dynamics of journal bearing rotor system in frequency domain . The rich dynamics of the journal bearing systems are discovered. The numerical illustrations of stable periodic motions are brought out with the initial conditions from analytical prediction.

<p> This study investigates the flow patterns and wall shear rate distributions downstream from a series of three glass model symmetric bifurcations, typical of the blood vessels in man. The models have a single included angle of 75° and total output to input flow area ratios of 0.75, 1.02 and 1.29, covering the physiological range. The Reynolds numbers studied (based on parent tube) were 400, 800 and 1200 in steady flow.</p> <p> Local fluid velocities were obtained at a number of axial positions along the bifurcation daughter tube via a neutrally buoyant tracer particle technique utilizing cine photography. This provided sufficient information to determine the three velocity components for each particle. The tangential and radial components were in general less than 6% of the mean axial velocity. In the case of the axial components, an analytical representation of the velocity in polar coordinates was obtained. This analytical function permits evaluation of wall shear rate distribution.</p> <p> The velocity pro£iles were found to be symmetric with respect to the plane of the bifurcation. At two diameters downstream from the carina the velocity profiles in the plane of the bifurcation showed a high peak near the inside wall of the branch. With distance downstream the peak was convected tangentially evening out the profile towards an axially symmetric mountain plateau with a dished top.</p> <p> Wall shear rate as a function of θ at constant axial position was represented by displaced cosine function. The highest shear rates always occurred on the inside wall of the daughter tube and the lowest on the outside wall. In general, the largest deviation from developed shear rates occurred close to the carina.</p> <p> The largest positive deviation in wall shear rate from developed values was found in the small area ratio bifurcation and the lowest wall shear rate value was found in the large area ratio bifurcation (a = 1.29) indicating possible flow separation near the carina. The biological implications of the shear rate information generated are discussed.</p> / Thesis / Master of Engineering (MEngr)

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The objective of this dissertation is to develop theoretical and computational tools for the study of qualitative changes in the dynamics of systems with discontinuities, also known as nonsmooth or hybrid dynamical systems, under parameter variations. Accordingly, this dissertation is divided into two parts. The analytical section of this dissertation discusses mathematical tools for the analysis of hybrid dynamical systems and their application to a series of model examples. Specifically, qualitative changes in the system dynamics from a nonimpacting to an impacting motion, referred to as grazing bifurcations, are studied in oscillators where the discontinuities are caused by impacts. Here, the study emphasizes the formulation of conditions for the persistence of a steady state motion in the immediate vicinity of periodic and quasiperiodic grazing trajectories in an impacting mechanical system. A local analysis based on the discontinuity-mapping approach is employed to derive a normal-form description of the dynamics near a grazing trajectory. Also, the results obtained using the discontinuity-mapping approach and direct numerical integration are found to be in good agreement. It is found that the instabilities caused by the presence of the square-root singularity in the normal-form description affect the grazing bifurcation scenario differently depending on the relative dimensionality of the state space and the steady state motion at the grazing contact. The computational section presents the structure and applications of a software program, TC-HAT, developed to study the bifurcation analysis of hybrid dynamical systems. Here, we present a general boundary value problem (BVP) approach to locate periodic trajectories corresponding to a hybrid dynamical system under parameter variations. A methodology to compute the eigenvalues of periodic trajectories when using the BVP formulation is illustrated using a model example. Finally, bifurcation analysis of four model hybrid dynamical systems is performed using TC-HAT. / Ph. D.

Dans cette thèse nous présentons une description théorique et numérique détaillée des instabilités et des dynamiques observées dans des systèmes quasi-unidimensionnels de particules en interaction répulsive soumises à un bain thermique. Lorsque le confinement transverse décroît, ces systèmes présentent une transition structurelle les faisant passer d'une configuration en ligne à une configuration en zigzag, homogène ou inhomogène. Nous avons mis en évidence et expliqué le changement de caractère de cette bifurcation qui passe de sur-critique à sous-critique. La description quantitative de configurations d'équilibre stables, appelées " bulles ", a été réalisée, celles-ci correspondent à une coexistence de domaines en ligne et en zigzag.La dynamique des " bulles " a été ensuite étudiée à l'aide d'un modèle de particule effective diffusant dans un potentiel périodique induit par le caractère discret du système. Lorsque plusieurs " bulles " coexistent, elles interagissent et se réorganisent pour former une configuration stable à une seule " bulle " selon des mécanismes de coalescence ou de collapse. Nous avons montré que la topologie de la configuration peut induire des effets de frustration conduisant à une interaction attractive ou répulsive selon les cas.Enfin, nous avons montré que les fluctuations transverses des particules divergent à l'approche des seuils de transition et expliqué ces comportements par l'apparition de modes mous dans le spectre de vibration. Cette description en modes propres nous a permis par ailleurs de comprendre l'augmentation observée de la diffusion d'une chaîne de particules dans un potentiel périodique asymétrique par rapport à une chaîne libre. / In this thesis, we provide a detailed theoretical and numerical study of instabilities and dynamics in quasi-one-dimensional systems of repulsively interacting particles in a thermal bath.When the transverse confinement decreases, theses systems display a structural transition from a line to an homogeneous or inhomogeneous staggered row configuration. We have exhibited and explained the supercritical or subcritical character of the bifurcation according to the particles interaction and to the system geometry. The quantitative description of stable equilibrium configurations called "bubbles" has been done, their shapes consist in coexistence of line and zigzag phases.The "bubble" dynamics has been modelized by considering an effective particle that diffuses in a periodic potential induced by the discrete character of the system. When several "bubbles" coexist, they interact and evolve towards a single stable "bubble" through coalescence and collapse mechanisms. We have shown that the configuration topology has to be taken into account and exhibited frustration effects leading to either an attractive or repulsive interaction between "bubbles". Then we have shown the divergence of the mean squared transverse displacements of the particles near the transition thresholds and analytically explained these critical behaviors by the existence of a soft mode in the configuration vibrational spectrum. With this eigenmodes description, we have also interpreted a diffusion enhancement of a particle file moving on an asymmetrical periodic potential with respect to the free file diffusion.

L'analyse des écoulements au sein des faisceaux d'assemblages constitue un volet important des études des réacteurs à eau pressurisée. Une mauvaise répartition thermique au sein de ces écoulements peut conduire à une crise d'ébullition nuisible à la sûreté du réacteur. De nombreuses études ont montré l'existence de phénomènes de réorganisation de structures aux grandes échelles dans ces écoulements. Cette thèse vise à améliorer notre compréhension de ces phénomènes, l'objectif étant de développer des modélisations aux petits échelles adaptées. Un travail bibliographique a mis en évidence les difficultés rencontrées par les simulations pour reproduire ces phénomènes, ainsi que de nombreux questionnements concernant leur caractère physique. Des simulations 3D ont été réalisées et ont permis d'identifier deux mécanismes de réorganisation pour les structures aux grandes échelles : un changement de signe de la vitesse transverse entre les crayons ou du tourbillon dans un sous-canal. Il est apparu qu'il semblait pertinent d'adopter l'hypothèse de Taylor pour considérer que les grandes structures 3D évoluaient comme un écoulement 2D transporté. Un gros volet de la thèse a concerné la mise en œuvre d'un code basé sur une méthode statistique pour un champ 2D dans le but de déterminer les états thermodynamiquement stables dans des géométries avec obstacles. Des similarités ont été obtenues entre les structures en REP et les états stables en 2D. Des simulations 2D ont permis d'identifier deux bifurcations possibles pour l'écoulement, qui présentent un parallèle avec les mécanismes de réorganisations 3D, et permettent ainsi de poser les bases d'une explication physique du phénomène / The analysis of fuel rod bundle flows constitute a key element of pressurized-water reactors safety studies. Indeed, an insufficient flow thermal mixing can lead to a boiling crisis, which is nefarious for the reactor safety. Numerous studies have shown the existence of reorganisation phenomena in the flow large-scale structures. This thesis work aims at improving our understanding of these phenomena, with the long-term goal of developing small-scales models suited for this type of flow. A bibliographic study has brought to light the challenges faced by simulations attempting to capture these phenomena, as well as various questions regarding their physical meaning. 3D simulations have been performed in order to study this flow ; they allowed to identify two reorganisation mechanisms for the large-scale structures consisting in a sign change for either a transverse velocity in rod-to-rod gaps or for a subchannel vortex. It appeared relevant to adopt a Taylor hypothesis in order to consider the evolution of large-scale 3D structures as transported-2D. A statistical method has then been applied to the 2D field in order to determine its thermodynamically-stable states in geometries with obstacles using the resolution of an optimization problem with a numerical calculation tool. Interesting similarities have been obtained between the PWR coherent structures and the stable states in a simplified 2D geometry. Further, 2D numerical simulations allowed to identify two different possible flow bifurcations. A parallel is drawn between these bifurcations and the two reorganizations observed in 3D simulations, laying the foundations for a physical explanation of this phenomenon

Determining the maximum power system loadability, as well as preventing the system from being operated close to the stability limits is very important in power systems planning and operation. The application of optimization techniques to power systems security and electricity markets is a rather relevant research area in power engineering. The study of optimization models to determine critical operating conditions of a power system to obtain secure power dispatches in an electricity market has gained particular attention. This thesis studies and develops optimization models and techniques to detect or avoid voltage instability points in a power system in the context of a competitive electricity market. A thorough analysis of an optimization model to determine the maximum power loadability points is first presented, demonstrating that a solution of this model corresponds to either Saddle-node Bifurcation (SNB) or Limit-induced Bifurcation (LIB) points of a power flow model. The analysis consists of showing that the transversality conditions that characterize these bifurcations can be derived from the optimality conditions at the solution of the optimization model. The study also includes a numerical comparison between the optimization and a continuation power flow method to show that these techniques converge to the same maximum loading point. It is shown that the optimization method is a very versatile technique to determine the maximum loading point, since it can be readily implemented and solved. Furthermore, this model is very flexible, as it can be reformulated to optimize different system parameters so that the loading margin is maximized. The Optimal Power Flow (OPF) problem with voltage stability (VS) constraints is a highly nonlinear optimization problem which demands robust and efficient solution techniques. Furthermore, the proper formulation of the VS constraints plays a significant role not only from the practical point of view, but also from the market/system perspective. Thus, a novel and practical OPF-based auction model is proposed that includes a VS constraint based on the singular value decomposition (SVD) of the power flow Jacobian. The newly developed model is tested using realistic systems of up to 1211 buses to demonstrate its practical application. The results show that the proposed model better represents power system security in the OPF and yields better market signals. Furthermore, the corresponding solution technique outperforms previous approaches for the same problem. Other solution techniques for this OPF problem are also investigated. One makes use of a cutting planes (CP) technique to handle the VS constraint using a primal-dual Interior-point Method (IPM) scheme. Another tries to reformulate the OPF and VS constraint as a semidefinite programming (SDP) problem, since SDP has proven to work well for certain power system optimization problems; however, it is demonstrated that this technique cannot be used to solve this particular optimization problem.