In this thesis, we study periodic solutions and stability of differential equations with piecewise constant argument of generalized type. These equations can be divided into three main classes: differential equations with retarded, alternately advanced-retarded, and state-dependent piecewise constant argument of generalized type. First, using the method of small parameter due to Poincar&eacute / , the existence and stability of periodic solutions of quasilinear differential equations with retarded piecewise constant argument of generalized type in noncritical case, that is, the unperturbed linear ordinary differential equation has not any nontrivial periodic solution, are investigated. The continuous and differential dependence of the solutions on an initial value and a parameter is considered. A new Gronwall-Bellmann type lemma is proved. Next, quasilinear differential equations with alternately advanced-retarded piecewise constant argument of generalized type is addressed. The critical case, when associated linear homogeneous system admits nontrivial periodic solutions, is considered. Using the technique of Poincar&eacute / -Malkin, criteria of existence of periodic solutions of such equations are obtained. One of the main auxiliary results is an analogue of Gronwall-Bellmann Lemma for functions with alternately advanced-retarded piecewise constant argument. Dependence of solutions on an initial value and a parameter is investigated. Finally, a new class of differential equations with state-dependent piecewise constant argument is introduced. It is an extension of systems with piecewise constant argument. Fundamental theoretical results for the equations: existence and uniqueness of solutions, the existence of the periodic solutions, the stability of the zero solution are obtained. Appropriate examples are constructed.

The research area of current PhD thesis is the analysis of time periodic Navier-Stokes equations in domains with cylindrical outlets to infinity. The objects of investigation is so called non-statonary Poiseuille solution in the straight cylinder and Navier-Stokes equations in system of cylinders. / Disertacijoje nagrinėjami Navjė-Stokso lygčių periodiniai laiko atžvilgiu uždaviniai srityse su cilindriniais išėjimais į begalybę. Pagrindiniai tyrimo objektai yra taip vadinami Puazelio sprendiniai tiesiame cilindre ir Stokso, bei Navjė-Stokso lygčių sistemos cilindrų sistemoje.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-27T15:55:51Z No. of bitstreams: 1 KELMEM DA CRUZ BARROSO - DISSERTAÇÃO PPGMAT 2011..pdf: 671604 bytes, checksum: ec714303d68155a38def404424574b39 (MD5) / Made available in DSpace on 2018-07-27T15:55:51Z (GMT). No. of bitstreams: 1 KELMEM DA CRUZ BARROSO - DISSERTAÇÃO PPGMAT 2011..pdf: 671604 bytes, checksum: ec714303d68155a38def404424574b39 (MD5) Previous issue date: 2011-09 / Capes / Para visualizar o resuma desta dissertação recomendamos o downloado do arquivo completo uma vez que o mesmo possui em sua estrutura fórmulas e sinais matemáticos que não foram possíveis transcrevê-los aqui. / To visualize the summary of this dissertation we recommend the downloado of the complete file since it has in its structure formulas and mathematical signs that were not possible to transcribe them here.

Provamos a existência de soluções periódicas de algumas equações diferenciais funcionais com retardamento sujeitas a condições de impulsos de auto-sustentação. Devido aos impulsos, soluções exibem descontinuidades de primeira espécie e isso força considerarmos espaços de fase mais gerais que C([-r,0],Rn). Mostramos que soluções periódicas podem emanar da origem através de bifurcações locais de Hopf. Também estabelecemos um teorema de existência de soluções periódicas lentamente espiralantes. Esse teorema é obtido combinando-se a condição de auto-sustentação com a ejetividade da origem em relação a um operador de retorno. / We prove the existence of periodic solutions of some retarded functional differential equations subjected to impulsive self-supporting conditions. Due to the impulses, solutions exhibit discontinuites of the first kind and this forces the consideration of more general phase spaces than C([-r,0],Rn). We show that periodic solutions can emanate from the origin through local Hopf bifurcations. We also state an existence theorem for slowly spiralling periodic solutions. This theorem is accomplished by a combination of the self-supporting condition with the ejectivity of the origin with respect to a return operator.

Submitted by Carolinne Stefane Souza (ssouza.carolinne@gmail.com) on 2018-02-23T20:46:35Z No. of bitstreams: 1 Dissertacao_Repositorio.pdf: 1968665 bytes, checksum: 81a4dfcb3e59ddb820eadef680510a59 (MD5) / Approved for entry into archive by Elza Mitiko Sato null (elzasato@ibilce.unesp.br) on 2018-02-26T17:20:13Z (GMT) No. of bitstreams: 1 souza_cs_me_sjrp.pdf: 1968665 bytes, checksum: 81a4dfcb3e59ddb820eadef680510a59 (MD5) / Made available in DSpace on 2018-02-26T17:20:13Z (GMT). No. of bitstreams: 1 souza_cs_me_sjrp.pdf: 1968665 bytes, checksum: 81a4dfcb3e59ddb820eadef680510a59 (MD5) Previous issue date: 2018-02-16 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Este trabalho tem como objetivo principal investigar condições que garantam a existência de soluções periódicas para certos tipos de equações diferenciais funcionais com retardamento e condições que garantam a permanência das soluções dessas equações. A teoria do grau coincidente será a principal ferramenta utilizada para obter os resultados referentes à existência de solução periódica. Por essa razão, uma atenção especial a essa teoria será dada nos primeiros capítulos. Resultados inéditos sobre permanência de soluções serão exibidos no último capítulo e ilustrados com exemplos numéricos. / The main objective of this work is to investigate conditions that guarantee the existence of periodic solutions to certain types of functional differential equations with delay and conditions that guarantee the permanence of the solutions of those equations.The coincidence degree theory will be the main tool used to obtain the results concerning the existence of periodic solution. For that reason, a special attention to that theory will be given in the first chapters. New results on the permanence of solutions will be shown in the last chapter and illustrated with numerical examples.

La motivation industrielle et mécanique du problème sera présentée pour un problème continu: élasticité linéaire avec une contrainte unilatérale. Un système masse-ressort avec un contact unilatéral en découle par discrétisation. Le but de cette thèse est d'étudier ces systèmes à vibro-impact de N degrés de liberté avec un contact unilatéral. Le système résultant est linéaire en l'absence de contact; Il est régi par une loi d'impact autrement. L'auteur identifie les modes non linéaires qui présentent une phase de contact collant pour un modèle à deux degrés de liberté en présence d'un obstacle rigide. L'application de premier retour de Poincaré est un outil fondamental pour étudier la dynamique près de solutions périodiques. Étant donné que la section de Poincaré est un sous-ensemble de l'interface de contact dans l'espace des phases, elle peut être tangente aux orbites pour les contacts rasants et conduire à une singularité en « racine carrée » déjà connue en Mécanique. Cette singularité est revisitée dans un cadre mathématique rigoureux. Elle implique la discontinuité du temps de premier retour. Enfin, l’instabilité des modes linéaire rasants est abordée. / The mechanical motivation is presented for a PDE with a constraint. The purpose of this thesis is to study N degree-of-freedom vibro-impact systems with an unilateral contact. The resulting system is linear in the absence of contact; it is governed by an impact law otherwise. The author identifies some nonlinear modes that display a sticking phase. The First Return Map is a fundamental tool to explore periodic solutions. Since the Poincaré section is a subset of the contact interface in the phase-space, it can be tangent to orbits which yields the well-known square-root singularity. This singularity is here revisited in a rigorous mathematical framework. Moreover, the study of this singularity implies a more important singularity: the discontinuity of the first return time. Finally, the square-root dynamics near the linear grazing modes which may lead to the instability of these linear grazing modes is studied.

Ces travaux s'articulent autour du calcul des solutions périodiques dans les systèmes dynamiques non linéaires, au moyen de méthodes numériques de continuation. La recherche de solutions périodiques se traduit par un problème avec conditions aux limites périodiques, pour lequel nous avons implémenté deux méthodes d'approximation : - Une méthode spectrale dans le domaine fréquentiel, l'équilibrage harmonique d'ordre élevé, qui repose sur une formulation quadratique des équations. Nous proposons en outre une extension de cette méthode aux cas de non-linéarités non rationnelles. - Une méthode pseudo-spectrale dans le domaine temporel, la collocation à l'aide fonctions polynômiales par morceaux. Ces méthodes transforment le problème continu en un système d'équations algébriques non linéaires, dont les solutions sont calculées par continuation à l'aide de la méthode asymptotique numérique. L'ensemble de ces outils, complétés d'une analyse linéaire de stabilité, sont intégrés au code de calcul MANLAB. Applications : Un modèle physique non-régulier de clarinette est étudié en détail : à partir de la branche de solutions statiques et ses bifurcations, on calcule les différentes branches de solutions périodiques, ainsi que leur stabilité et leurs bifurcations. Ce modèle est ensuite adapté au cas du saxophone, pour lequel on intègre une caractérisation acoustique expérimentale, afin de mieux tenir compte de la géométrie complexe de l'instrument. Enfin, nous étudions un modèle physique simplifié de violon, avec une non-régularité liée frottement de Coulomb. / Periodic solutions of nonlinear dynamical systems are the focus of this work. We compute periodic solutions through a BVP formulation, solved with two numerical methods: - a spectral method, in the frequency domain: the hogh-order Harmonic Balance Method, using a quadratic formulation of the original equations. We also propose an extension to nonrational nonlinearities. - a pseudo-spectral method, in the time domain : the arthogonal collocation at Gauss point, with piece-wise polynomial interpolation. Both methods lead to a system of nonlinear algebraic equations, and its solutions are computed by a continuation algorithm : the Asymptotic Numerical Method. These methods are embeded in the numerical package MANLAB, together with a linear stability analysis. Application We then apply these methods to physical models of several instruments : a clarinet, a saxophone, and a violin. The clarinet model contains a non-smooth contact between the reed and the mouthpiece. The study focuses on the evolution of frequency, loudness, and spectrum along the branch of periodic solutions when varying the mouth pressure. The saxophone model is very similar, but an experimental characterization of the bore is used in that case. Finally, the violin model with a non-smooth Coulomb contact law and a simplified resonator is studied, showing the variety of models that can be treated using this method.

No description available.

L'objectif de cette thèse est l'étude qualitative de certains modèles de la dynamique et la distribution des bactéries dans une rivière. Il s'agit de la stabilité des états stationnaires et l'existence des solutions périodiques. Nous considérons, dans la première partie de la thèse, un système d'équations différentielles ordinaires qui modélise les interactions et la dynamique de quatre espèces de bactéries dans une rivière. Nous avons étudié le comportement asymptotique des états stationnaires. L'étude de la stabilité des états stationnaires est essentiellement faite par la construction d'une fonction de Lyapunov combinée avec le principe d'invariance de LaSalle. D'autre part, l'existence des solutions périodiques est démontrée en utilisant le théorème de continuation de Mawhin. La deuxième partie de la thèse est consacrée à l'étude d'un système de convection-diffusion non-autonome. Ce modèle tient compte du transport des bactéries. Nous étudions l'analyse qualitative des solutions, nous déterminons l'ensemble limite du système et nous démontrons l'existence des états stationnaires positifs. L'étude de l'existence des états stationnaires (les seuls qu'il soit possible d'obtenir) est basée sur le théorème de Leray-Schauder. / The objective of this thesis is the qualitative study of some models of the dynamic and the distribution of bacteria in a river. We are interested in the stability of equilibria and the existence of periodic solutions. The thesis can be divided into two parts; the first part is concerned with a mathematical analysis of a system of differential equations modelling the dynamics and the interactions of four species of bacteria in a river. The asymptotic behavior of equilibria is established. The stability study of equilibrium states is mainly done by construction of Lyapunov functions combined with LaSalle's invariance principle. On the other hand, the existence of periodic solutions is proved under certain conditions using the continuation theorem of Mawhin. In the second part of this thesis, we propose a non-autonomous convection-reaction diffusion system with nonlinear reaction source functions. This model refers to the quantification and the distribution of antibiotic resistant bacteria (ARB) in a river. Our main contributions are : (i) the determination of the limit set of the system; it is shown that it is reduced to the solutions of the associated elliptic system; (ii) sufficient conditions for the existence of a positive solution of the associated elliptic system based on the Leray Schauder's degree theory.

Orientador: Lucas Catão de Freitas Ferreira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-27T04:26:31Z (GMT). No. of bitstreams: 1 CastanedaCenturion_NestorFelipe_D.pdf: 1063498 bytes, checksum: bbaaad01ffead1389f469e88505aada5 (MD5) Previous issue date: 2015 / Resumo: Por apresentar basicamente fórmulas, o Resumo, na íntegra, poderá ser visualizado no texto completo da tese digital / Abstract: The complete Abstract is available with the full electronic document / Doutorado / Matematica / Doutor em Matemática