Nesta tese apresentamos uma contribuição para o estudo da transição do retrato de fase de uma equação diferencial descontínua específica ao longo de uma linha de descontinuidade. A equação diferencial que tratamos neste trabalho é a das linhas de curvatura principal de uma superfície S contendo uma curva distinguida B e imersa em R^3. A linha de descontinuidade é a curva B, a qual é o bordo comum de duas superfícies suaves justapostas que formam S. Na primeira parte do trabalho consideramos a superfície seccionalmente suave, S = S+ U B U S-, obtida pela justaposição de S+ e S- ao longo do bordo comum B. O estudo da configuração principal de S nos casos em que as linhas de curvatura principal das superfícies S+ e S- tem contato quadrático ou cruzam transversalmente B foi feito por comparação com a configuração principal de uma superfície suave, obtida de S pelo processo da \"regularização\" ao longo da curva de descontinuidade B. Na segunda parte do trabalho estudamos as linhas de curvatura principal de uma superfície S em R^3 com bordo B e da superfície suave obtida de S através dos processos de engrossamento e regularização definidos por Garcia e Sotomayor em [5], onde os autores consideraram o caso genérico, sem pontos umbílicos e contato quadrático de uma linha de curvatura principal com B. Damos aqui continuidade ao estudo feito em [5] analisando o caso de contato cúbico com o bordo B. Obtivemos que dos pontos da curva bordo comum B de contato quadrático e de cruzamento transversal emergem, sobre a superfície regularizada, pontos umbílicos Darbouxianos dos tipos D1 e D3, enquanto que, para o ponto sobre B de contato cúbico obtivemos pontos umbílicos Darbouxianos dos tipos D1, D2 e D3 e também pontos umbílicos não Darbouxianos dos tipos D12 e D23. [5] Garcia, R., and Sotomayor, J. Umbilic and tangential singularities on configurations of principal curvature lines. Anais da Academia Brasileira de Ciências 74, 1 (2002), 117. / In this work we present a contribution to the study of the transition of the phase portrait of a specific discontinuous differential equation along a line of discontinuity. The differential equations under consideration will be that of the principal curvature lines of a surface S with a distinguished curve B immersed in R^3, where the line of discontinuity is the curve B which is the common border of two smooth surfaces attached to make up S. In the first part of the work we consider a piecewise smooth surface S = S+ U B U S-, obtained by the juxtaposition of two smooth surfaces S+ and S- along their common border B. The analysis of the principal configuration of S in the cases where the principal curvature lines of the surfaces S+ and S- have quadratic contact or cross transversally B was carried out by comparison with a smooth surface, obtained from S by the \"regularization\" along the discontinuity curve B. In the second part of the work we study the principal curvature lines of a surface S in R^3 with boundary B and of the smooth surface obtained from S by thickening and smoothing introduced by Garcia and Sotomayor in [5], where they considered the generic case of no umbilic points and at most quadratic contact of principal lines with B. Here we pursue the study in [5] and analyze the case of cubic contact with the border B. We established that while from quadratic contact points with B emerge on the smoothed surface Darbouxian umbilics of D1 and D3 types, from the cubic contact points appear Darbouxian umbilics of types D1, D2 and D3 as well as non Darbouxian points of types D12 and D23. [5] Garcia, R., and Sotomayor, J. Umbilic and tangential singularities on configurations of principal curvature lines. Anais da Academia Brasileira de Ciências 74, 1 (2002), 117.

We study a Dirichlet problem, investigating existence and uniqueness for semipositone and superlinear nonlinearities. We make use of Pohozaev identities, energy arguments, and bifurcation from a simple eigenvalue.

The main subject of this thesis is solitons due to degenerate parametric four-wave mixing. Derivation of the governing equations is carried out for both spatial solitons (slab waveguide) and temporal solitons (optical fibre). Higher-order effects that are ignored in the standard paraxial approximation are discussed and estimated. Detailed analysis of conventional solitons is carried out. This includes discovery of various solitons families, linear stability analysis of fundamental and higher-order solitons, development of theory describing nonlinear dynamics of higher-order solitons. The major findings related to the stationary problem are bifurcation of a two-frequency soliton family from an asymptotic family of infinitely separated one-frequency solitons, jump bifurcation and violation of the bound state principle. Linear stability analysis shows a rich variety of internal modes of the fundamental solitons and existence of a stability window for higher-order solitons. Theory for nonlinear dynamics of higher-order solitons successfully predicts the position and size of the stability window, and various instability scenarios. Equivalence between direct asymptotic approach and invariant based approach is demonstrated. A general analytic approach for description of localised solutions that are in resonance with linear waves (quasi-solitons and embedded solitons) is given. This includes normal form theory and approximation of interacting particles. The main results are an expression for the amplitude of the radiating tail of a quasi-soliton, and a two-fold criterion for existence of embedded solitons. Influence of nonparaxiality on soliton stability is investigated. Stationary instability threshold is derived. The major results are shift and decreasing of the size of the stability window for higher-order solitons. The latter is the first demonstration of the destabilizing influence of nonparaxiality on higher-order solitons. Analysis of different aspects of solitons is based on universal approaches and methods. This includes Hamiltonian formalism, consideration of symmetry properties of the model, development of asymptotic models, construction of perturbation theory, application of general theorems etc. Thus, the results obtained can be extended beyond the particular model of degenerate four-wave mixing. All theoretical predictions are in good agreement with the results of direct numerical modeling.

Notre premier objectif dans ce travail est de donner une démonstration du changement<br />de la stabilité de la branche supercritique de solutions périodiques bifurquées<br />dans le cadre des équations diérentielles à retard, en se basant sur les deux étapes<br />suivantes:<br />(i) Réduction de l'équation à un système en dimension deux par la formule de variation<br />de la constante et le théorème de la variété centre.<br />(ii) Estimation de la distance entre la solution de l'équation initiale et la solution pé-<br />riodique bifurquée.<br />Nous obtenons ainsi un domaine de stabilité de la branche supercritique.<br />Le second objectif est d'étudier une équation différentielle à un seul retard issue<br />d'un modèle en dynamique de population cellulaire sanguine (Haematopoiese).<br />Ce modèle, initialement introduit par Mackey (1978) présente une position d'équilibre<br />triviale qui est instable et une famille de positions d'équilibre non triviales dont la<br />stabilité dépend du retard.<br />Nous montrons l'existence d'une valeur critique ¿0 du retard \tau autour de laquelle nous<br />obtenons un changement de stabilité de cette famille de positions d'équilibre en fonction<br />du retard.<br />Nous avons ainsi introduit un modèle approché en fonction de cette valeur critique du<br />retard qui coincide avec celui de Mackey pour la valeur du retard \tau = \tau_{0}. Le modèle<br />approché possède un point d'équilibre trivial et un non trivial ne dépendant pas du<br />retard.<br />Par une étude du modèle approché analogue à celle du modèle de Mackey, nous obtenons<br />en particulier l'existence d'une branche de solutions périodiques bifurquées à<br />partir du point d'équilibre non trivial. Enn nous donnons un algorithme explicite de<br />calcul des éléments de la bifurcation.

<p>While alternans in a single cardiac cell appears through a simple</p><p>period-doubling bifurcation, in extended tissue the exact nature</p><p>of the bifurcation is unclear. In particular, the phase of</p><p>alternans can exhibit wave-like spatial dependence, either</p><p>stationary or traveling, which is known as <italic>discordant</italic></p><p>alternans. We study these phenomena in simple cardiac models</p><p>through a modulation equation proposed by Echebarria-Karma. In</p><p>this dissertation, we perform bifurcation analysis for their</p><p>modulation equation.</p><p>Suppose we have a cardiac fiber of length l, which is</p><p>stimulated periodically at its x=0 end. When the pacing period</p><p>(basic cycle length) B is below some critical value B_c,</p><p>alternans emerges along the cable. Let a(x,n) be the amplitude</p><p>of the alternans along the fiber corresponding to the n-th</p><p>stimulus. Echebarria and Karma suppose that a(x,n) varies</p><p>slowly in time and it can be regarded as a time-continuous</p><p>function a(x,t). They derive a weakly nonlinear modulation</p><p>equation for the evolution of a(x,t) under some approximation,</p><p>which after nondimensionization is as follows: </p><p> &partial_t a = σ a + <bold>L</bold> a - g a <super>3</super>,</p><p>where the linear operator</p><p> <bold>L</bold> a = &partial_xxa - &partial_x a -Λ<super>-1</super> ∫ <super>0</super> _x a(x',t)dx'.</p><p>In the equation, σ is dimensionless and proportional to</p><p>B_c - B, i.e. σ indicates how rapid the pacing is,</p><p>Λ<super>-1</super> is related to the conduction velocity (CV) of the</p><p>propagation and the nonlinear term -ga<super>3</super> limits growth after the</p><p>onset of linear instability. No flux boundary conditions are</p><p>imposed on both ends.</p><p>The zero solution of their equation may lose stability, as the</p><p>pacing rate is increased. To study the bifurcation, we calculate</p><p>the spectrum of operator <bold>L</bold>. We find that the</p><p>bifurcation may be Hopf or steady-state. Which bifurcation occurs</p><p>first depends on parameters in the equation, and for one critical</p><p>case both modes bifurcate together at a degenerate (codimension 2)</p><p>bifurcation.</p><p>For parameters close to the degenerate case, we investigate the</p><p>competition between modes, both numerically and analytically. We</p><p>find that at sufficiently rapid pacing (but assuming a 1:1</p><p>response is maintained), steady patterns always emerge as the only</p><p>stable solution. However, in the parameter range where Hopf</p><p>bifurcation occurs first, the evolution from periodic solution</p><p>(just after the bifurcation) to the eventual standing wave</p><p>solution occurs through an interesting series of secondary</p><p>bifurcations.</p><p>We also find that for some extreme range of parameters, the</p><p>modulation equation also includes chaotic solutions. Chaotic waves</p><p>in recent years has been regarded to be closely related with</p><p>dreadful cardiac arrhythmia. Proceeding work illustrated some</p><p>chaotic phenomena in two- or three-dimensional space, for instance</p><p>spiral and scroll waves. We show the existence of chaotic waves in</p><p>one dimension by the Echebarria-Karma modulation equation for</p><p>cardiac alternans. This new discovery may provide a different</p><p>mechanism accounting for the instabilities in cardiac dynamics.</p> / Dissertation

Recently the phenomena of Bose-Einstein condensates have been observed in laboratories, and the related problems are extensively studied. In this paper we consider the nonlinear Schrödinger equation in the laser beam rotating magnetic field and compute its corresponding energy functional under the mass conservative condition. By separating time and space variables, factoring real part and image part, and discretizing via finite difference method, the original equation can be transformed to a large scale parametrized polynomial systems. We use continuation method to find the solutions that satisfy the mass conservative condition. We will also explore bifurcation points on the curves and other solutions lying on bifurcation branches. The numerical results show that when the rotating angular momentum is small, we can find the solutions by continuation method along some particular curves and these curves are regular. As the angular momentum is increasing, there will be more bifurcation points on curves.

Lienard Equations serve as the elegant models for oscillating circuits. Motivated by this fact, this thesis addresses the stability property of a class of delayed Lienard equations. It shows the existence of the Hopf bifurcation around the steady state. It has both practical and theoretical importance in determining the criticality of the Hopf bifurcation. For such purpose, center manifold analysis on the bifurcation line is required. This thesis uses operator differential equation formulation to reduce the infinite dimensional delayed Lienard equation onto a two-dimensional manifold on the critical bifurcation line. Based on the reduced two-dimensional system, the so called Poincare-Lyapunov constant is analytically determined, which determines the criticality of the Hopf bifurcation. Numerics based on a Matlab bifurcation toolbox (DDE-Biftool) and Matlab solver (DDE-23) are given to compare with the theoretical calculation. Two examples are given to illustrate the method.

The manufacturing process has been a heavily studied area over the past century. The study completed herein has established a foundation for the future of manufacturing research. The next step of this industry is to become proficient at the micro and nano scale levels of manufacturing. In order to accomplish this goal, the modeling of machining system needs to be completely understood throughout the entire process. In effort to attack this problem, this study will focus on the boundaries present in machining systems; and will define and interpret the associated phenomena. This particular focus is selected since nearly all manufacturing related studies concentrate on continuous processes; which by definition considers only one particular operation. There is a need to understand the phenomena corresponding to interactions of multiple processes of manufacturing systems. As a means to this end, the nonlinear phenomena associated in the continuous domains of machining systems will be modeled as linear to ensure the boundary interactions are clearly observed. Interference of additional nonlinearities is not the focus of this research. In this dissertation, the mechanical model for a widely accepted machine-tool system is presented. The state and continuous domains are defined with respect to the boundaries in this system (contact and frictional force acting at the point of tool and work-piece contact). The switching sets defining plane boundaries for the continuous systems of this machine-tool will be defined and studied herein. The forces and force products, at the point of switching from one continuous system to another, govern the pass-ability of the machine-tool through the respective boundary. The forces and force product components at the switching points are derived according to discontinuous systems theory Luo [1]. Mapping definitions and notations are developed through the switching sets for each of the boundaries. A mapping structure and notation for periodic interrupted cutting, non-cutting and chip seizure motions are defined. The interruption of the chip flow for a machining system will be investigated through a range of system parameters. The prediction of interrupted periodic cutting, non-cutting and chip seizure motion will be completed via closed form solutions for this machine-tool. The state of this system is defined to utilize the theory of Luo [1]. This is necessary to properly handle the frictional force boundary at the chip/tool interface, the onset of cutting boundary and the contact boundary between the tool and work-pieces. The predictions by this method will be verified via numerical simulation and comparison to existing research. A goal of this research is to illustrate the effects of the dynamical systems interacting at the frictional force (chip/tool) boundary and the chip onset of growth and vanishing boundary. The parameter space for this machine-tool model is studied through numerical and analytical predictions, which provide limits on the existence of interrupted periodic cutting, non-cutting and chip seizure motions.

Texte remanié de: Th. Ph. D.--Rijksuniversiteit Groningen, 1999. / Publ. à partir de la thèse soutenue par Gerton Lunter sous la dir. de Henk Broer et Gert Vegter. Bibliogr. p. [159]-165. Index.

Thèse doctorat : Génie Civil : Villeurbanne, INSA : 2004. / Titre provenant de l'écran-titre. Bibliogr. p. 258-266.