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.

Using satellite and aerial imagery, the planform of a large number of bifurcations is gathered and interpreted in terms of quantifiable parameters. The majority of the observed bifurcations displays a conspicuous asymmetry in terms of bifurcation angle, upstream channel curvature, length of the downstream bifurcates. The first part of this work focuses on the effect of the differences in channel width between the upstream channel and the bifurcates added together and between the bifurcates themselves. Therefore, two parameters are derived, namely the bifurcation enlarge- ment that accounts for the primer factor and the channel width asymmetry that accounts for the latter factor. Within the data considered, the two derived factors are poorly correlated and each factor is also poorly correlated to other planform and morphodynamic parameters. Especially the lack of correlation between the observed bifurcation enlargement and channel width asymmetry indicates that the bifurcates do not obey a regime equilibrium configuration by which the channel carrying more water and sediment is also larger. The second part then deals with the mechanisms underlying the equilibrium configurations of bifurcations, especially how the channel width differences affect these configurations, using a well-known simplifying physical relationship that couples the distributions of water and sediment discharges with the planimetry and the bathymetry of the bifurcation node. The two planform param- eters appear to compete against each other in terms of equilibrium stability. The bifurcation enlargement associated with a shallower and steeper downstream flow destabilizes the bifurcation as the flow condition in both anabranches is closer to the critical sediment motion condition. On the other hand, the channel width asymmetry is associated with an increase of the transport capacity in the narrower dominated channel. Finally, the stability threshold of the observed bifurcation is computed for each observed bifurcation, showing that the bifurcation planform notably affects the stability of the configuration in natural settings.

The convective fluid dynamics and chaos between two parallel plates with temperature discrepancy has been investigated via classic and extended Lorenz system. Both the classic 3-dimensional and extended 5-dimensional Lorenz system are developed by truncating a double Fourier series, which is the solution of the streamline function. Boundary conditions are also considered. The implicit discrete mapping method has been employed to solve the classic and extended Lorenz system, and the motion stability is determined by the eigenvalue analysis. Bifurcation diagram varying with Rayleigh parameter and Prandtl parameter are obtained by solving the stable and unstable period-m motions (m=1,2,4). Symmetric period-1 to asymmetric period-4 motions have been illustrated in the phase space. Therefore, the route from period-1 to period-4 motions to chaos through the period-doubling bifurcation has been demonstrated in the classic and extended Lorenz system. For the extended 5-dimensional Lorenz system, the harmonic frequency-amplitude characteristics are also presented, which provides energy distribution in the parameter space. On bifurcation tree, the non-spiral and spiral homoclinic orbits have been seen and been illustrated in 2-D view and 3-D view. Such homoclinic orbits represent the asymptotic convection steady state that generates the chaos in the convective fluid dynamics. The rich dynamical behaviors of the convective fluid are discovered, and this investigation may help one understand the chaotic dynamics for other thermal convection problems.

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.