Motivated by models from evolutionary population dynamics, we study a general class of nonlinear difference equations called matrix models. Under the assumption that the projection matrix is non-negative and irreducible, we prove a theorem that establishes the global existence of a continuum with positive equilibria that bifurcates from an extinction equilibrium at a value of a model parameter at which the extinction equilibrium destabilizes. We give criteria for the global shape of the continuum, including local direction of bifurcation and its relationship to the local stability of the bifurcating positive equilibria. We discuss a relationship between backward bifurcations and Allee effects. Illustrative examples are given

Dans ce mémoire, nous étudions un système prédateur-proie de Gause généralisé avec une récolte de proie contrôlée et une fonction de réponse de Holling de type III généralisée. Nous introduisons une fonction de récolte contrôlée sur les proies tenant compte du nombre de proies et dépendant d'un seuil de récolte. Ceci permet de rendre le système réaliste, d'optimiser la récolte, et de prévenir la possibilité d'extinction des espèces que le système avec récolte constante pouvait avoir pour toutes valeurs de paramètres. Ce type de fonction de récolte implique a priori la manipulation d'un système discontinu: nous étudions donc des techniques de lissage de ces discontinuités par régularisation. Nous faisons d'abord un retour sur les systèmes sans et avec récolte de proie constante en traçant les diagrammes de bifurcations exacts et les portraits de phase de ces systèmes. Ensuite, nous étudions le système discontinu et les méthodes de régularisation afin de choisir la plus optimale. Finalement, nous assemblons le tout avec l'étude du système avec récolte de proie régularisé, en passant par l'étude complète du système avec approvisionnement de proie, et donnons les différents effets sur les portraits de phase selon les conditions initiales. / In this master thesis, we study a generalized Gause predator-prey system with controlled prey harvest and a generalized Holling response function of type III. We introduce a controlled prey harvesting function taking into account the number of preys with a harvesting threshold. This makes the system realistic, it optimizes the harvesting, and it prevents the possibility of species' extinction which exists in the system with constant harvest for all parameters. This type of harvesting function a priori implies handling a discontinuous system : therefore we study smoothing techniques of such discontinuities by regularization. We first return on systems without and with constant harvest by drawing the exact bifurcation diagrams and phase portraits of those systems. Then, we study the discontinuous system and the regularization methods in order to choose the optimal one. Finally, we put together everything by studying the regularized prey harvesting system through a complete study of the prey stocking system, and we highlight the different effects on the phase portraits under the initial conditions.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Thesis advisor: Andrzej Herczynski / The magnetic pendulum investigated in this experiment closely models various forms of the gravitational pendulum. However, the apparatus used in this experiment allows for greater insight as the constant and periodic forces can be easily varied. This project extends the previous work of Sang-Yoon Kim and Francis Moon on the magnetic pendulum by including an additional degree of freedom. This additional degree of freedom allows for a greater understanding of the bifurcation points observed. / Thesis (BS) — Boston College, 2007. / Submitted to: Boston College. College of Arts and Sciences. / Discipline: Physics. / Discipline: College Honors Program.

Seja M um conjunto compacto e conexo do plano que seja a união dos subconjuntos conexos N e S. Seja Z_L=(X_L,Y_L) uma família a um parâmetro de campos vetoriais descontínuos, onde X_L está definida em N e Y_L em S. Ambos os campos X_L e Y_L, assim como as suas dependências em L, são suaves i. e. de classe C^\\infty; a descontinuidade acontece na fronteira comum entre N e S. O objetivo deste trabalho é estudar as bifurcações que ocorrem em certas famílias de campos vetoriais descontínuos seguindo as convenções de Filippov. Aplicando o método da regularização, introduzido por Sotomayor e Teixeira e posteriormente aprofundado por Sotomayor e Machado à família de campos vetoriais descontínuos Z_L obtemos uma família de campos vetoriais suaves que é próxima da família descontínua original. Usamos esta técnica de regularização para estudar, por comparação com os resultados clássicos da teoria suave, as bifurcações que ocorrem nas famílias de campos vetoriais descontínuos. Na literatura há uma lista de bifurcações de codimensão um, no contexto de Filippov, apresentada mais completamente, no artigo de Yu. A. Kuznetsov, A. Gragnani e S. Rinaldi, One-Parameter Bifurcations in Planar Filippov Systems, Int. Journal of Bifurcation and Chaos, vol. 13, No. 8: 2157--2188, (2003). Alguns dos casos dessa lista já eram conhecidos por Kozlova, Filippov e Machado. Neste trabalho nos propomos a estudar as bifurcações de alguns dos casos, apresentados no artigo de Kuznetsov et. al, através do método da regularização dessas famílias. Nesta Tese consubstanciamos matematicamente a seguinte conclusão: As bifurcações das famílias descontínuas analisadas ficam completamente conhecidas através das bifurcações apresentadas pelas respectivas famílias regularizadas, usando recursos da teoria clássica suave. / Let M be a connected and compact set of the plane which is the union of the connected subsets N and S. Let Z_L=(X_L,Y_L) be a one-parameter family of discontinuous vector fields, where X_L is defined on N and Y_L on S. The two fields X_L, Y_L and their dependences on L are smooths, i. e., are of C^\\infty class; the discontinuity happens in the common boundary of N and S. The objective of this work is to study the bifurcations which occurs in certains families of discontinuous vector fields following the conventions of Filippov. Applying the regularization method, introduced by Sotomayor and Teixeira, to the family of discontinuous vector fields Z_L we obtain a family of regular vector fields which is close to the original family of discontinuous vector fields. In the literature there is a list of codimension one bifurcation, in the Filippov sense, presented more completely, in the article of Yu. A. Kuznetsov, A. Gragnani e S. Rinaldi, One-Parameter Bifurcations in Planar Filippov Systems, Int. Journal of Bifurcation and Chaos, vol. 13, No. 8: 2157--2188, (2003). Some of those cases was already known by Kozlova, Filippov and Machado. In this work we propose to study the bifurcations of some of those cases, presented in the article of Kuznetsov et. al, by the method of regularization of those families. In this thesis we justify mathematically the following conclusion: The bifurcations of the analysed discontinuous families are completelly known by the bifurcations contained in the respective regularized families, using the methods of the classical theory of regular vector fields.

Seja M um conjunto compacto e conexo do plano que seja a união dos subconjuntos conexos N e S. Seja Z_L=(X_L,Y_L) uma família a um parâmetro de campos vetoriais descontínuos, onde X_L está definida em N e Y_L em S. Ambos os campos X_L e Y_L, assim como as suas dependências em L, são suaves i. e. de classe C^\\infty; a descontinuidade acontece na fronteira comum entre N e S. O objetivo deste trabalho é estudar as bifurcações que ocorrem em certas famílias de campos vetoriais descontínuos seguindo as convenções de Filippov. Aplicando o método da regularização, introduzido por Sotomayor e Teixeira e posteriormente aprofundado por Sotomayor e Machado à família de campos vetoriais descontínuos Z_L obtemos uma família de campos vetoriais suaves que é próxima da família descontínua original. Usamos esta técnica de regularização para estudar, por comparação com os resultados clássicos da teoria suave, as bifurcações que ocorrem nas famílias de campos vetoriais descontínuos. Na literatura há uma lista de bifurcações de codimensão um, no contexto de Filippov, apresentada mais completamente, no artigo de Yu. A. Kuznetsov, A. Gragnani e S. Rinaldi, One-Parameter Bifurcations in Planar Filippov Systems, Int. Journal of Bifurcation and Chaos, vol. 13, No. 8: 2157--2188, (2003). Alguns dos casos dessa lista já eram conhecidos por Kozlova, Filippov e Machado. Neste trabalho nos propomos a estudar as bifurcações de alguns dos casos, apresentados no artigo de Kuznetsov et. al, através do método da regularização dessas famílias. Nesta Tese consubstanciamos matematicamente a seguinte conclusão: As bifurcações das famílias descontínuas analisadas ficam completamente conhecidas através das bifurcações apresentadas pelas respectivas famílias regularizadas, usando recursos da teoria clássica suave. / Let M be a connected and compact set of the plane which is the union of the connected subsets N and S. Let Z_L=(X_L,Y_L) be a one-parameter family of discontinuous vector fields, where X_L is defined on N and Y_L on S. The two fields X_L, Y_L and their dependences on L are smooths, i. e., are of C^\\infty class; the discontinuity happens in the common boundary of N and S. The objective of this work is to study the bifurcations which occurs in certains families of discontinuous vector fields following the conventions of Filippov. Applying the regularization method, introduced by Sotomayor and Teixeira, to the family of discontinuous vector fields Z_L we obtain a family of regular vector fields which is close to the original family of discontinuous vector fields. In the literature there is a list of codimension one bifurcation, in the Filippov sense, presented more completely, in the article of Yu. A. Kuznetsov, A. Gragnani e S. Rinaldi, One-Parameter Bifurcations in Planar Filippov Systems, Int. Journal of Bifurcation and Chaos, vol. 13, No. 8: 2157--2188, (2003). Some of those cases was already known by Kozlova, Filippov and Machado. In this work we propose to study the bifurcations of some of those cases, presented in the article of Kuznetsov et. al, by the method of regularization of those families. In this thesis we justify mathematically the following conclusion: The bifurcations of the analysed discontinuous families are completelly known by the bifurcations contained in the respective regularized families, using the methods of the classical theory of regular vector fields.

by Luk Wai Shing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1993. / Includes bibliographical references (leaves 44-45). / ABSTRACT --- p.i / ACKNOWLEDGMENTS --- p.ii / Chapter CHAPTER 1. --- INTRODUCTION --- p.1 / Chapter CHAPTER 2. --- DRAWBACKS OF CLASSICAL THEORY --- p.4 / Chapter 2.1 --- Review of Sequential Iterative Methods --- p.4 / Chapter 2.2 --- Visualization Techniques --- p.8 / Chapter 2.3 --- Review of Deflation --- p.10 / Chapter CHAPTER 3. --- THE IMPROVEMENT OF THE ABERTH METHOD --- p.11 / Chapter 3.1 --- The Durand-Kerner method and the Aberth method --- p.11 / Chapter 3.2 --- The generalized Aberth method --- p.13 / Chapter 3.3 --- The modified Aberth Method for multiple-zero --- p.13 / Chapter 3.4 --- Choosing the initial approximations --- p.15 / Chapter 3.5 --- Multiplicity estimation --- p.16 / Chapter CHAPTER 4. --- THE HIGHER-ORDER ITERATIVE METHODS --- p.18 / Chapter 4.1 --- Introduction --- p.18 / Chapter 4.2 --- Convergence analysis --- p.20 / Chapter 4.3 --- Numerical Results --- p.28 / Chapter CHAPTER 5. --- PARALLEL DEFLATION --- p.32 / Chapter 5.1 --- The Algorithm --- p.32 / Chapter 5.2 --- The Problem of Zero Component --- p.34 / Chapter 5.3 --- The Problem of Round-off Error --- p.35 / Chapter CHAPTER 6. --- HOMOTOPY ALGORITHM --- p.36 / Chapter 6.1 --- Introduction --- p.36 / Chapter 6.2 --- Choosing Q(z) --- p.37 / Chapter 6.3 --- The arclength continuation method --- p.38 / Chapter 6.4 --- The bifurcation problem --- p.40 / Chapter 6.5 --- The suggested improvement --- p.41 / Chapter CHAPTER 7. --- CONCLUSION --- p.42 / REFERENCES --- p.44 / APPENDIX A. PROGRAM LISTING --- p.A-l / APPENDIX B. COLOR PLATES --- p.B-l

by Kwok, Wong-chuen Tony. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1993. / Includes bibliographical references (leaves [85-86]). / Acknowledgement / Abstract / Chapter I --- Laguerre's Method in Polynomial Zero-finding / Chapter 1 --- Background --- p.1 / Chapter 2 --- Introduction and Problems of Laguerre´ةs Method --- p.3 / Chapter 2.1 --- Laguerre´ةs Method in Symmetrie-Cluster Problem / Chapter 2.2 --- Cyclic Behaviour / Chapter 2.3 --- Supercluster Problem / Chapter 3 --- Proposed Enhancement to Laguerre 's Method --- p.9 / Chapter 3.1 --- Analysis of Adding a Zero or Pole / Chapter 3.2 --- Proposed Algorithm / Chapter 4 --- Conclusion --- p.17 / Chapter II --- Homotopy Methods applied to Polynomial Zero-finding / Chapter 1 --- Introduction --- p.18 / Chapter 2 --- Overcoming Bifurcation --- p.22 / Chapter 3 --- Comparison of Homotopy Algorithms --- p.27 / Chapter 4 --- Conclusion --- p.29 / Appendices / Chapter I --- Laguerre's Method in Polynomial Zero-finding / Chapter 0 --- Naming of Testing Polynomials / Chapter 1 --- Finding All Zeros using Proposed Laguerre's Method / Chapter 2 --- Experiments: Selected Pictures of Comparison of Proposed Strategy with Other Strategy / Chapter 3 --- Experiments: Tables of Comparison of Proposed Strategy with Other Strategy / Chapter 4 --- Distance Colorations and Target Colorations / Chapter II --- Homotopy Methods applied to Polynomial Zero-finding / Chapter 1 --- Comparison of Algorithms using Homotopy Method / Chapter 2 --- Experiments: Selected Pictorial Comparison / Chapter III --- An Example Demonstrating Effect of Round-off Errors References

We study Schrödinger operators perturbed by non-compact (spatially extended) defects. We consider two models: a one-dimensional (1D) dimer structure with a global phase shift, and a two-dimensional (2D) honeycomb structure with a line-defect or "edge''. In both the 1D and 2D settings, the non-compact defects are modeled by adiabatic, domain wall modulations of the respective dimer and honeycomb structures. Our main results relate to the rigorous construction of states via bifurcations from continuous spectra. These bifurcations are controlled by asymptotic effective (homogenized) equations that underlie the protected or non-protected character of the states. In 1D, the states we construct are localized solutions. In 2D, they are "edge states'' - time-harmonic solutions which are propagating (plane-wave-like) parallel to a line-defect or "edge'' and are localized transverse to it. The states are described as protected if they persist in the presence of spatially localized (even strong) deformations of the global phase defect (in 1D) or edge (in 2D). The protected states bifurcate from "Dirac points'' (linear/conical spectral band-crossings) in the continuous spectra and are seeded by an effective Dirac equation. The (more conventional) non-protected states bifurcate from spectral band edges are seeded by an effective Schrödinger equation. Our 2D model captures many aspects of the phenomenon of topologically protected edge states observed in honeycomb structures such as graphene and "artificial graphene''. The protected states we construct in our 1D dimer model can be realized as highly robust TM- electromagnetic modes for a class of photonic waveguides with a phase-defect. We present a detailed computational study of an experimentally realizable photonic waveguide array structure.

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.