Thesis (M.Sc. (Dept. of Mathematics, Faculty of Natural Sciences))--University of the Western Cape, 2009. / Bibliography: leaves 180-195.

Thèse (M.Sc.)--Université Laval, 2005. / Titre de l'écran-titre (visionné le 15 décembre 2005). Bibliogr.

Dans une perspective de report modal, la performance adaptative est un facteur important du développement des systèmes de transports collectifs. Il est donc fondamental de disposer d’outils permettant d’évaluer leur vulnérabilité à des perturbations et le rôle de certaines actions comme l’information des voyageurs. Le modèle PERTURB et la plate-forme de simulation, développés dans le cadre de cette recherche ont ainsi pour objet de mesurer la vulnérabilité structurelle et fonctionnelle des transports en commun ainsi que le rôle de l’information des voyageurs dans la régulation de situations perturbées. Trois champs théoriques sont mobilisés : théorie de systèmes, théorie des graphes et systèmes multi-agents. Les simulations sont réalisées sur un terrain d’étude (Ile-de-France) et des systèmes de transport théoriques. Elles permettent d’effectuer des préconisations en matière d’Aménagement des transports et d’information des voyageurs mais suscitent également de nouvelles interrogations. Comment intégrer l’information des voyageurs dans l’optimisation du traitement des situations perturbées ? Comment concilier une transparence de l’information et éviter des reports trop massifs de flux de voyageurs ? / IIn a perspective of modal transfer, adaptative performance is an important factor in the development of public transport systems. Tools are therefore necessary to allow estimation of their vulnerability to perturbations, as well as that of the role played by specific actions like passengers’ information. The PERTURB model and the simulation platform developed in this research aim to measure the structural and functional vulnerability of public transport, as well as the role played by passengers’ information in the regulation of disrupted situations. This study explores three theoretical fields : systems’ theory, graphs theory and multi-agents systems. Simulations are used in a field study (the region of Ile-de-France) and in theoretical transportation systems. These simulations allow us not only to formulate recommendations, regarding transport planning and passengers’ information, but they also rise new issues. How can passengers’ information be integrated in order to get the optimal management of disrupted situations ? Is it possible to deliver clear information while avoiding excessive transfer of passengers’ flow at the same time ?

La thèse concerne l'étude des cycles limites d'une équation différentielle sur le plan (la deuxième partie du 16ème problème de Hilbert). La notion de "cycle limite" a une grande importance dans la théorie de la stabilité, elle est introduite par Poincaré vers la fin du 19ème siècle et désigne une orbite périodique isolée. Le but de cette thèse est : d'établir l'existence d'une borne supérieure finie, pour le nombre de cycle limites d'une équation quadratique dans le plan. Ce problème est aussi appelé 16ème problème d' Hilbert infinitésimal. Probablement, l'outil le plus fondamental pour l'étude de la stabilité et les bifurcations des orbites périodiques est l'application de Poincaré, défini par Henri Poincaré en 1881. Cependant, la méthode de Melnikov nous donne une excellente procédure pour déterminer le nombre de cycles limites dans une bande continue de cycles qui sont préservés sous perturbation. En effet, le nombre, les positions et les multiplicités des équations différentielles planes perturbées avec une petite perturbation non nulle sont déterminées par le nombre, les positions et les multiplicités des zéros des fonctions génératrices. La fonction de Melnikov est plus précisément, appelé la fonction de Melnikov de premier- ordre. Si cette fonction est identiquement nulle à travers la bande continue de cycles, on calcule ce qu'on appelle " la fonction de Melnikov d'ordre supérieure ". Ensuite, une analyse d'ordre supérieure est nécessaire, ce qui peut être fait par " l'algorithme de Françoise. Les discussions et les calculs présentés dans notre travail sont limités non seulement à la fonction de Melnikov de premier ordre, mais aussi pour les fonctions de Melnikov de deuxième -ordre. Ces outils seront utiles pour résoudre notre problématique. Les activités de recherche menées dans le cadre de cette recherche sont divisées en quatre parties : La première partie de cette thèse, traite les systèmes dynamiques plans et l'existence de cycles limites. Nous souhaitons après résoudre le problème suivant: Calculer la cyclicité de l'oscillateur asymétrique perturbé de Duffing. Dans la deuxième partie, nous sommes intéressés de la cyclicité à l'extérieur de l'anneau périodique de l'oscillateur de Duffing pour une perturbation particulière, puis, nous fournissons un diagramme de bifurcation complet pour le nombre de zéros de la fonction de Melnikov associée dans un domaine complexe approprié en se basant sur le principe de l'argument. Le nombre de cette cyclicité est égal à trois. Dans la troisième partie, nous étudions la cyclicité à l'intérieur ainsi que à l'extérieur de double boucle homocline pour une perturbation cubique arbitraire de l'oscillateur de Duffing en utilisant les mêmes techniques de Iliev et Gavrilov dans le cas d'un Hamiltonien asymétrique de degré quatre. Notre principal résultat est que deux au plus cycle limite peuvent bifurquer de la double homocline. D'autre part, il est représenté, qu'après bifurcation de eight-loop un cycle limite étranger est née, qui ne soit pas contrôlée par un zéro lié par les intégrales Abéliennes, ce cycle supplémentaire est appelé " Alien ". / This thesis concerns the study of limit cycles of a differential equation in the plane (The second part of the 16th Hilbert problem). The concept of "limit cycle" has a great importance in the theory of stability; Poincaré introduces this notion at the end of the 19th century and denotes an isolated periodic orbit. The purpose of this thesis: Find an upper bound finite to the number of limit cycles of a quadratic equation in the plane. This problem is so- called the infinitesimal Hilbert 16th problem. Probably, the most basic tool for studying the stability and bifurcations of periodic orbits is the Poincaré, defined by Henri Poincaré in 1881. However, Melnikov's method gives us an excellent method for determining the number of limit Cycles in a continuous band of cycles that are preserved under perturbation. In fact, the number, positions and multiplicities of perturbed planar differential equations for a small nonzero parameters, are determined by the number, positions and multiplicities of the zeros of the generating functions. The Melnikov function is more precisely, called the first-order Melnikov function. If this function is identically equal zero across the continuous band of cycles, one computes the so-called "Higher order Melnikov function". Then, a higher order analysis is necessary which can be done by making use of the so called "the algorithm of Françoise". The discussions and computation presented in this thesis are restricted not only to the first order Melnikov function, but also to the second-order Melnikov functions. These tools will be useful to resolve the question problem. The research activities in the framework of this thesis are divided into four parts: The first part of this thesis, discusses planar dynamical systems and the existence of limit cycles. We wish to solve the following problem: Calculate the cyclicity of the perturbed asymmetric oscillator Duffing. In the second part, we are interested of the cyclicity of the exterior period annulus of the asymmetrically perturbed Duffing oscillator for a particular perturbation, then, we provide a complete bifurcation diagram for the number of zeros of the associated Melnikov function in a suitable complex domain based on the argument principle. The number of this cyclicity is equal to three. In the third part, we study the cyclicity of the interior and exterior eight-loop especially for arbitrary cubic perturbations by using the same techniques of Iliev and Gavrilov in the case of an asymmetric Hamiltonian of degree four. Our main result is that at most two limit cycles can bifurcate from double homoclinic loop. On the other hand, it is appears after bifurcation of eight-loop an "Alien" limit was born, which is not covered by a zero of the related Abelian integrals.

Estudos de perturbações em sistemas gravitacionais no âmbito da Relatividade Geral vêm sofrendo grandes desenvolvimentos nos últimos anos, especialmente em face da evolução dos modernos detectores de ondas gravitacionais. Abordamos neste trabalho as perturbações de diferentes cenários. Principiamos com a métrica de Vaidya, utilizada para descrever espaços-tempos esfericamente simétricos e dependentes do tempo. Nossas simulações mostraram que as freqüências dos modos quasi-normais (MQN\'s) apresentam um novo efeito inercial para variações rápidas da função de massa, retornando depois ao comportamento adiabático. Em seguida, apresentamos um modelo para a evaporação de mini buracos negros por radiação de Hawking inspirado no cenário de criação destes objetos em aceleradores de partículas, previsto pelas novas teorias com dimensões extras. Nosso modelo, baseado na métrica de Vaidya n-dimensional, tornou possível a análise de MQN\'s resultando na possibilidade de se obter os parâmetros relevantes do buraco negro, como a sua massa inicial e o número de dimensões extras, a partir de medições experimentais. Finalmente, realizamos um estudo sobre uma nova solução denominada gravastar, proposta como um modelo alternativo para o estágio final de estrelas com grande massa. Obtivemos limites para os parâmetros da solução e verificamos a sua estabilidade frente a perturbações axiais, concluindo positivamente a respeito da possibilidade de se distinguir entre buracos negros e gravastares com base no seu espectro de MQN\'s. / Perturbative studies of gravitational systems in General Relativity have gone through big developments in the last years, especially due to the evolution of the modern gravitational wave detectors. We consider in this work different perturbations in different scenarios. Firstly we consider the Vaidya metric, mainly used to describe time-dependent spherically symmetric spacetimes. Our simulations show that the frequencies of the quasinormal modes (QNM\'s) present a new inertial effect for rapidly varying mass functions, returning afterwards to the adiabatic behavior. Next we present a model for evaporating mini black holes in particle accelerators, in the context of the new gravity models with extra dimensions. With our model, based on the n-dimensional Vaidya metric, we are able to perform a QNM analysis which results in the possibility of obtaining the parameters of the black hole, such as its initial mass and the number of extra dimensions, from the experimental measurements. Finally, we present a study of a new solution, the gravastar, proposed as an alternative model for the end state of massive stars. We obtain bounds for the parameters of the solution and verify its stability against axial perturbations. Our results indicate that the gravastar\'s QNM spectrum can indeed be used to distinguish a black hole from a gravastar.

Magister Scientiae - MSc / This thesis is concerned with singularly perturbed two-parameter problems. We study a tted nite difference method as applied on two different meshes namely a piecewise mesh (of Shishkin type) and a graded mesh (of Bakhvalov type) as well as a tted operator nite di erence method. We notice that results on Bakhvalov mesh are better than those on Shishkin mesh. However, piecewise uniform meshes provide a simpler platform for analysis and computations. Fitted operator methods are even simpler in these regards due to the ease of operating on uniform meshes. Richardson extrapolation is applied on one of the tted mesh nite di erence method (those based on Shishkin mesh) as well as on the tted operator nite di erence method in order to improve the accuracy and/or the order of convergence. This is our main contribution to this eld and in fact we have achieved very good results after extrapolation on the tted operator finitete difference method. Extensive numerical computations are carried out on to confirm the theoretical results. / South Africa

This work presents a combination of analytical and numerical approaches to gain an insight of the dynamics of marine risers and micro machined resonators. Despite their scale difference, we show that both systems share similarities in terms of initial static deformation, quadratic and cubic nonlinearities, and internal resonances. First, we utilize the state space method to study the eigenvalue problem of vertical riser. An orthonormalization step is introduced to recover the numerical scheme during numerical integration and we investigate the effect of applied tension, apparent weight, and higher-order modes on the accuracy of the scheme. We show that the method is advantageous to find eigenvalues and mode shapes of vertical risers in comparison to other methods. The work is extended to study the eigenvalue problem of inclined risers considering the influence of static deflection, self-weight and mid-plane stretching. The linear dynamics is solved using Galerkin method. The results demonstrate that under the influence of tension and configuration angle, the frequencies exhibit commensurate ratio with respect to the first natural frequency leading to the possible activation of internal resonances. Next, we study the nonlinear interactions of inclined risers considering two-to-one and three-to-one internal resonances under single and multifrequency excitations. The multiple times scale method is applied to study the nonlinear interaction and results are compared to those from a Galerkin solution showing good agreement. Time histories and perturbation’s response curves, in addition to the dynamical solution obtained by Galerkin and stability analysis using Floquet theory are utilized to examine the system. These results feature nonlinear energy exchange, saddle node jumps, and Hopf bifurcations leading to complex dynamic motion that can endanger the riser structure. Finally, the analysis using pertubation is extended to investigate the two-to-one internal resonance in micromachined arch beams between its first two symmetric modes. The response is analyzed using the perturbation method considering the nonlinear interaction and two simultaneous excitations at higher AC voltages. Good agreement is found among the results of pertubations, Galerkin and experimental data from fabricated Silicon arch beam. Different types of bifurcations are observed which can lead to quasi-periodic and potentially chaotic motions.

Introduction: Idiopathic Parkinson’s Disease is a multisystem neurodegenerative disease that is characterized by asymmetric impairment in regions of the midbrain, forebrain, and brainstem. Of the known neurodegenerative diseases, Parkinson’s is the second most commonly diagnosed worldwide with a global prevalence expected to reach 9 million individuals by 2030. As fall rates range between 35-68% annually, falling during walking is amongst the primary concerns for this demographic. Interestingly, despite the close association between loss of arm swing (due to Parkinson’s Disease) and future falls, evidence to-date has not examined the effect different arm swing conditions have on walking stability during unperturbed and perturbed (cognitive and mechanical) conditions. Dynamic stability research in this demographic is further limited in that evidence examining differences between the least and most affected leg is sparse. Research Objectives: To examine the differences between natural arm swing (unrestricted) and when arm swing was physical restricted (restricted) in people with Parkinson’s Disease. The effect of arm swing was assessed when people with Parkinson’s Disease walked in steady-state, dual-task, destabilizing terrains as well as in response to slips. Additionally, this thesis examined differences between the least and most affected sides, during the aforementioned conditions, that stem from the asymmetric progression in Parkinson’s Disease. Methods: Twenty individuals with Parkinson’s Disease were recruited for this research. Individuals walked on a CAREN-Extended System with unrestricted (natural) and restricted (absent) arm swing. Arm conditions were combined with steady-state walking, walking while performing a secondary dual-task, walking on minor destabilizing environments (hilly, rocky and mediolateral translational), and in response to slips for the heel-strikes of the perturbed (slipped) leg and recovery (contralateral) leg. The minor destabilizing terrains were assessed separately to steady-state walking for the arm swing condition resulting in three types of analyses (arms-rocky, arms-rolling hills, and arms-mediolateral). Data were processed in Vicon, Visual 3D, and OpenSim before being exported to Matlab to calculate dynamic stability (Margin of Stability, Harmonic Ratios and Coefficient of Variation), average spatiotemporal parameters, as well as trunk linear and angular velocities. Statistical analyses were conducted in SPSS with a significance level set a priori at (p<0.05). Results: During unperturbed walking with the restricted arm swing condition, compared to unrestricted, average trunk angular velocity increased in the transverse plane while instantaneous linear velocity at heel-strike decreased in the sagittal plane. Further, on the least affected leg, the Margin of Stability increased, average step length decreased, and coefficient of variation for step length increased. Contrastingly, step time coefficient of variation increased in the most affected leg. In the presence of the dual-task, average angular velocity in the frontal plane increased, average step time decreased (most affected leg), and step width coefficient of variation increased (bilaterally). Compared to unrestricted arm swing, restricted arm swing reduced average step length (arm-rolling hills) and time (arm-rocky), and increased COV step time (arm-rolling hills). The arm-rolling hills analysis revealed that the most affected leg had a shorter step length than the least affected. The destabilizing surface effects revealed that during the arm-rolling hills and arm-rocky analyses step time decreased, step width increased, and the COV for step time, length and width increased. No main effects occurred for the arm-mediolateral analysis. Additionally, when comparing the arm swing conditions in response to a slip, the restricted arm swing condition, compared to unrestricted, caused a faster step time during the slipped step. Compared to the most affected leg, the least affected had a wider step width during the slipped step. During the recovery step, the least affected leg had a larger anteroposterior Margin of Stability and longer step time than the most affected. Conclusion: The findings revealed that when people with Parkinson’s Disease walk without arm swing, trunk rotational velocity increases which internally perturbs gait. This destabilization elicited unique responses from dynamic stability metrics that were specific to the terrain encountered. Since Parkinson’s Disease primarily affects movement timing, the results suggest that loss of arm swing is particularly perturbing to foot placement timing while changes in spatial foot placement reflect compensation to maintain an existing level of global dynamic stability and symmetry. Additionally, the evidence indicates that the independent behavior of the least and most affected leg respond uniquely to loss of arm swing. However, as people with Parkinson’s Disease adjust the least affected leg’s foot placement to mirror the contralateral leg, functional interlimb differences may only be revealed when individuals encounter perturbations.

This dissertation reports high-resolution experimental study and numerical analysis of the rubidium dimer 31Πg, "6" ^"1" "Σ" _"g" ^"+" , "3" ^"3" "Π" _"g" , and "4" ^"3" "Σ" _"g" ^"+" excited electronic states. The term energies of over 2 400 observed ro-vibrational levels spanning a large range of rotational and vibrational quantum numbers were measured with the perturbation facilitated optical-optical double resonance technique 24 000 cm-1 – 26 000 cm-1 above the ground state minimum of Rb2. The excited electronic states were probed by exciting Rb2 molecules from the thermally populated ro-vibrational levels of the "X" ^" 1" "Σ" _"g" ^"+" ground electronic state through intermediate levels of the mixed" " "A" ^"1" "Σ" _"u" ^"+" " ~ " "b" ^"3" "Π" _"u" electronic states. Probe laser resonance was detected by measuring the laser induced fluorescence from the excited electronic states to the "a" ^"3" "Σ" _"u" ^"+" triplet ground state. The ro-vibrational term energies from each electronic state were fit to molecular constants using the Dunham expansion. These molecular constants were subsequently used to generate Rydberg-Klein-Rees model potential energy functions. The spin multiplicity of the electronic states as well as the vibrational numbering of the triplet electronic states were determined by resolving the bound-free emission from the excited ro-vibrational levels to the triplet ground state. / Physics

The method of singular-perturbation analysis is applied to the determination of range-fuel-time optimal aircraft trajectories. The problem is shown to break down into three sub-problems which are studied separately. In particular, the inner layer containing the altitude path-angle dynamics is analyzed in detail. The outer solutions are discussed in an earlier work. As a step forward in solving the ensuing nonlinear two-point boundary-value problem, linearization of the equations is suggested. Conditions for the stability of the linearized boundary-layer equations are discussed. Also, the question of parameter selection to fit the solution to the split boundary conditions is resolved. Generation of feedback laws for the angle-of-attack from the linear analysis is discussed. Finally, the techniques discussed are applied to a numerical example of a missile. The linearized feedback solution is compared to the exact solution obtained using a multiple shooting method. / Master of Science