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.

Orientador: Claudio Aguinaldo Buzzi / Banca: Marco Antônio Teixeira / Banca: Paulo Ricardo da Silva / Resumo: O objetivo principal desse trabalho é provar, sob certas hipóteses de transversalidade e sobre os autovalores, que se uma família a um-parâmetro de equações diferenciais possuindo, para um determinado valor do parâmetro, um laço homoclínico conectado a um ponto de equilíbrio do tipo sela, então existe uma variedade central invariante, de dimensão dois, que contém o laçco homoclínico, que contém todas as trajetórias que permanecem numa vizinhança do laço homoclínico e ainda é tangente ao autoespaço gerado por autovetores associados aos autovalores que determinam o laço homoclínico. / Abstract: The main goal of this work is to prove, under certain hypothesis of transversality and about the eigenvalues, that if a one-parameter family of ordinary differential equations possess, for a determined value of the parameter, a homoclinic loop connected to an equilibrium point of type saddle, then there exists an invariant center manifold, of dimension two, that contains the homoclinic loop, that contains all trajectories which stay in a small neighborhood of the homoclinic loop and that is tangent to the eigenspace spanned by the eigenvectors associated to the eigenvalues that determine the homoclinic loop. / Mestre

Imagine fluid moving through a long pipe or channel, and we inject dye or solute into this pipe. What happens to the dye concentration after a long time? Initially, the dye just moves along downstream with the fluid. However, it is also slowly diffusing down the pipe and towards the edges as well. It turns out that after a long time, the combined effect of transport via the fluid and this slow diffusion results in what is effectively a much more rapid diffusion process, lengthwise down the stream. If 0 <nu << 1 is the slow diffusion coeffcient, then the effective longitudinal diffusion coeffcient is inversely proportional to 1/nu, i.e. much larger. This phenomenon is called Taylor Dispersion, first studied by GI Taylor in the 1950s, and studied subsequently by many authors since, such as Aris, Chatwin, Smith, Roberts, and others. However, none of the approaches used in the past seem to have been mathematically rigorous. I'll propose a dynamical systems explanation of this phenomenon: specifically, I'll explain how one can use a Center Manifold reduction to obtain Taylor Dispersion as the dominant term in the long-time limit, and also explain how this Center Manifold can be used to provide any finite number of correction terms to Taylor Dispersion as well.

Nous avons étudié un système différentiel régi par deux dynamiques : l'une de type variété centrale et l'autre de type oscillation rapide périodique. Nous avons cherché à obtenir des informations sur le comportement qualitatif du système et à l'approcher. Nous avons démontré l'existence d'une dynamique asymptotique rapidement oscillante et nous l'avons utilisée pour approcher le système. Ensuite, nous avons appliqué ces résultats à l'étude d'un système écologique d'interaction proie-prédateur. De plus, nous avons utilisé la théorie des B-séries pour obtenir des développements formels à tout ordre des quantités liées à la dynamique asymptotique. Enfin, nous avons approché le système pour tout temps par la composée d'un changement de variable et de la solution d'un système différentiel partiellement découplé. / We have studied a differential system ruled by two dynamics : a center manifold dynamics and a periodic highly oscillating dynamics. We wanted to find informations about the qualitative behaviour of the system, and to approximate it. We have proved the existence of a highly oscillating asymptotic dynamics, and we have used it approximate the system. Then, we have applied this results to an ecological system of prey-predator interaction. Moreover, we have used the B-series theory to obtain formal expansions of the quantities related to the center manifold. Lastly, we have approximated the system for all time by the composition of a change of variable and of the solution of a partially decoupled differential system.

Mathematical models displaying travelling waves appear in a variety of domains. These waves are often faced with different kinds of perturbations. In some cases, these perturbations result in propagation failure, also known as wave-blocking. Wave-blocking has been studied in the case of several specific models, often with the help of numerical tools. In this thesis, we will display a technique that uses symmetry and a center manifold reduction to find a criterion which defines regions in parameter space where a wave will be blocked. We focus on waves with low velocity and small symmetry-breaking perturbations, which is where the blocking initiates; the organising center. The range of the tools used makes the technique easily generalizable to higher dimensions. In order to demonstrate this technique, we apply it to the bistable equation. This allows us to do calculations explicitly. As a result, we show that wave-blocking occurs inside a wedge originating from the organising center and derive an expression for this wedge to leading order. We verify our results with some numerical simulations.

Mathematical models displaying travelling waves appear in a variety of domains. These waves are often faced with different kinds of perturbations. In some cases, these perturbations result in propagation failure, also known as wave-blocking. Wave-blocking has been studied in the case of several specific models, often with the help of numerical tools. In this thesis, we will display a technique that uses symmetry and a center manifold reduction to find a criterion which defines regions in parameter space where a wave will be blocked. We focus on waves with low velocity and small symmetry-breaking perturbations, which is where the blocking initiates; the organising center. The range of the tools used makes the technique easily generalizable to higher dimensions. In order to demonstrate this technique, we apply it to the bistable equation. This allows us to do calculations explicitly. As a result, we show that wave-blocking occurs inside a wedge originating from the organising center and derive an expression for this wedge to leading order. We verify our results with some numerical simulations.

In this thesis, we introduce two new classes of differential equations, which essentially extend, in several directions, impulsive differential equations and equations on time scales. Basics of the theory for quasilinear systems are discussed, and particular results are obtained so that further investigations of the theory are guaranteed. Applications of the newly-introduced systems are shown through a center manifold theorem, and further, Hopf bifurcation Theorem is proved for a three-dimensional discontinuous dynamical system.

Mathematical models displaying travelling waves appear in a variety of domains. These waves are often faced with different kinds of perturbations. In some cases, these perturbations result in propagation failure, also known as wave-blocking. Wave-blocking has been studied in the case of several specific models, often with the help of numerical tools. In this thesis, we will display a technique that uses symmetry and a center manifold reduction to find a criterion which defines regions in parameter space where a wave will be blocked. We focus on waves with low velocity and small symmetry-breaking perturbations, which is where the blocking initiates; the organising center. The range of the tools used makes the technique easily generalizable to higher dimensions. In order to demonstrate this technique, we apply it to the bistable equation. This allows us to do calculations explicitly. As a result, we show that wave-blocking occurs inside a wedge originating from the organising center and derive an expression for this wedge to leading order. We verify our results with some numerical simulations.

Orientador: Edson Denis Leonel / Resumo: Este estudo objetiva provar que sistemas dinâmicos de dimensão N, de codimensão um e satisfazendo as condições do teorema da bifurcação de Hopf, podem ser expressos em uma forma analítica simplificada que preserva a topologia do espaço de fases da configuração original, na vizinhança do ponto de equilíbrio. A esta forma simplificada é atribuído o nome de forma normal. Para tanto, foi utilizado a teoria da variedade central, necessária para reduzir a dimensão de sistemas à sua variedade bidimensional, e o teorema das formas normais, utilizando-se como método para determinar a forma simplificada da variedade central associada aos sistemas dinâmicos, atendendo as condições do teorema da bifurcação de Hopf. A partir da análise dos resultados aqui encontrados foi possível construir a prova matemática de que sistemas de dimensão N, atendendo as condições do teorema de Hopf, podem ser reescritos em uma expressão analítica geral e simplificada. Enfim, através deste estudo foi possível resumir todos os resultados aqui obtidos em um teorema geral que, além de reduzir a custosa tarefa de obtenção de formas normais, abrange sistemas N-dimensionais com ocorrência da bifurcação de Hopf. / Abstract: In this work we prove the following: consider a N-dimensional system that is reduced to its center manifold. If it is proved the system satisfies the conditions of Hopf bifurcation theorem, then the original system of differential equations is rewritten in a simpler analytical expression that preserves the phase space topology. This last is also known as the normal form. The center manifold is used to derive a reduced order expression, and the normal form theory is applied to simplify the form of the dynamics on the center manifold. The key results here allow constructing a general mathematical proof for the normal form of N-dimensional systems reduced to its center manifold. In the class of dynamical systems under Hopf bifurcations, the present work reduces the work done to obtain normal forms. / Mestre

Made available in DSpace on 2014-06-11T19:30:22Z (GMT). No. of bitstreams: 0 Previous issue date: 2006Bitstream added on 2014-06-13T20:07:43Z : No. of bitstreams: 1 carnevarollojr_rp_me_sjrp.pdf: 404426 bytes, checksum: 4ea777e4e28a9a139f9b0bcb3f7b0f7c (MD5) / O objetivo principal desse trabalho é provar, sob certas hipóteses de transversalidade e sobre os autovalores, que se uma família a um-parâmetro de equações diferenciais possuindo, para um determinado valor do parâmetro, um laço homoclínico conectado a um ponto de equilíbrio do tipo sela, então existe uma variedade central invariante, de dimensão dois, que contém o laçco homoclínico, que contém todas as trajetórias que permanecem numa vizinhança do laço homoclínico e ainda é tangente ao autoespaço gerado por autovetores associados aos autovalores que determinam o laço homoclínico. / The main goal of this work is to prove, under certain hypothesis of transversality and about the eigenvalues, that if a one-parameter family of ordinary differential equations possess, for a determined value of the parameter, a homoclinic loop connected to an equilibrium point of type saddle, then there exists an invariant center manifold, of dimension two, that contains the homoclinic loop, that contains all trajectories which stay in a small neighborhood of the homoclinic loop and that is tangent to the eigenspace spanned by the eigenvectors associated to the eigenvalues that determine the homoclinic loop.