Dissertations / Theses on the topic 'Planar vector field'

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Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)
Neste trabalho temos como objetivo estudar o número e a distribuição de ciclos limites em sistemas diferenciais lineares por partes. Em particular estudamos o número de ciclos limites do sistema diferencial linear por partes planar ˙x = −y − ε φ ( x) , ˙y = x, onde ε 6= 0 é um parâmetro pequeno e φ é uma função periódica linear por partes ímpar de período 4 . Provamos que dado um inteiro arbitário positivo n, o sistema acima possui exatamente n ciclos limites na faixa |x| ≤ 2 (n + 1 ). Consequentemente, existem sistemas diferenciais lineares por partes contendo uma infinidade de ciclos limites no plano real. Inicialmente obtemos uma quota inferior par a o número destes ciclos limites na faixa | x| ≤ 2 (n + 1 ) via Teoria do Averaging . Em seguida , utilizando a Teoria de Campos de Vetores Rodados, verificamos que o sistema acima tem exatamente n ciclos limites na faixa | x| ≤ 2 (n + 1 )
The main goal of this work aim to study the number and distribution of limit cycles in piecewise linear differential systems. In particular we consider the planar piecewise linear differential system ˙x = −y − ε φ ( x) , ˙y = x, where ε 6= 0 is a small parameter and φ is an odd piecewise linear periodic function of period 4 . We prove that given an arbitrary positive integer n, the system above has exactly n limit cycles in the strip | x| ≤ 2 (n + 1 ) . Consequently, there are piecewise differential systems containing an infinite number of limit cycles in the real plane. First we get a lower bound on the number of limit cycles in the strip |x| ≤ 2 (n + 1 ) via Averaging Theory. In the following , using the Theory of Rotated Vector Fields, we see that above system has exactly n limit cycles in the strip | x| ≤ 2 (n + 1 )

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Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)
Consideramos dois casos principais de bifurcação de órbitas periódicas não hiperbólicas que dão origem a ciclos limite. Nosso estudo é feito para sistemas lineares por partes com três zonas em sua fórmula mais geral, que inclui situações sem simetria. Obtemos estimativas tanto para a amplitude como para o período do ciclo limite e apresentamos uma aplicação de interesse em engenharia: sistemas de controle.
We consider two main cases of bifurcation of non hyperbolic periodic orbits that give rise to limit cycles. Our study is done concerning piecewise linear systems with three zones in the more general formula that includes situations without symmetry. We obtain estimates for both the amplitude and the period of limit cycles and we present a applications of interest in engineering: control systems.

Orientador: Paulo Ricardo da Silva
Banca: Weber Flavio Pereira
Banca: Marcelo Messias
Resumo: Consideramos dois casos principais de bifurcação de órbitas periódicas não hiperbólicas que dão origem a ciclos limite. Nosso estudo é feito para sistemas lineares por partes com três zonas em sua fórmula mais geral, que inclui situações sem simetria. Obtemos estimativas tanto para a amplitude como para o período do ciclo limite e apresentamos uma aplicação de interesse em engenharia: sistemas de controle.
Abstract: We consider two main cases of bifurcation of non hyperbolic periodic orbits that give rise to limit cycles. Our study is done concerning piecewise linear systems with three zones in the more general formula that includes situations without symmetry. We obtain estimates for both the amplitude and the period of limit cycles and we present a applications of interest in engineering: control systems.
Mestre

Aquesta tesi es situa en el marc de la teoriaqualitativadelssistemesd’equacionsdiferencials en el pla. Cada capítol conté un aspectediferent, però en totsells es tractenproblemes, la soluciódelsqualsestà basada en el rol que hi juguen les simetriesdiscretes i continues (reversibilitat o simetries de Lie) de campsvectorialsplans. A la introducció, es dóna un resumdelsresultatsmésconeguts i s’hiintrodueix la notació que es fa servir al llarg de la tesi. En el segon i tercer capítol, s’aborda el problema de trobarl’expressió explícita del canvilinealitzant o orbitalmentlinealitzantd#un camp vectorial suau a partir del coneixementd’uncommutador, en el cas de la linealització, o una simetria de Lie, en el cas de la linealització orbital. Cada capítol finalitzaambexemplesil.lustratius del procedimentconstructiudelscanvis. Al Capítol 5 s’apliquenelsresultatsdelscapítolsanteriors, combinatsamblinealitzacionsDarbouxianes. Concretament, es considera un sistema quadràtictipusLotka-Volterra i es caracteritzen les selles linealitzables i orbitalmentlinealitzablesmitjançant la troballadelscanvislinealitzants o orbitalmentlinealitzants. En el sisè capítol, s’utilitzal’existènciad’unàlgebra de simetriespuntuals de Lie per donar informació sobre l’existència i localitzaciód’òrbitesperiòdiques. En particular, quanl’àlgebra de simetriespuntuals de Lie d’unaequació diferencial escalar de segónordreautònoma i suau té dimensiómajor o igual a dos, definim les anomenadesfuncionsfonamentals que enspermeten estudiar les òrbitesperiòdiques al pla de fases. En el cas particular d’equacionspolinomials de Liénard, mostrem la no existència de cicles límit en tot el pla de fases. Finalment, al Capítol 7 es relacionen elssistemes reversibles amb el problema del centre aixícomamb el problema de la integrabilitat analítica. Consideremsistemesd’equacionsdiferencialsanalíticsamb centres degenerats i mostrem que poden transformar-se, després d’un reescalat del temps, en un sistema lineal i reversible. El coneixement de integralsprimeresens proporciona un procediment per caracteritzar, en alguns casos, la condició de reversibilitat del centre degenerat. D’altra banda, relacioneml’existència de integralsprimeresanalítiquesamb la reversibilitat orbital analítica en el cas de singularitatsdèbils no degenerades.

Aquesta tesi es situa en el marc de la teoriaqualitativadelssistemesd’equacionsdiferencials en el pla. Cada capítol conté un aspectediferent, però en totsells es tractenproblemes, la soluciódelsqualsestà basada en el rol que hi juguen les simetriesdiscretes i continues (reversibilitat o simetries de Lie) de campsvectorialsplans. A la introducció, es dóna un resumdelsresultatsmésconeguts i s’hiintrodueix la notació que es fa servir al llarg de la tesi. En el segon i tercer capítol, s’aborda el problema de trobarl’expressió explícita del canvilinealitzant o orbitalmentlinealitzantd#un camp vectorial suau a partir del coneixementd’uncommutador, en el cas de la linealització, o una simetria de Lie, en el cas de la linealització orbital. Cada capítol finalitzaambexemplesil.lustratius del procedimentconstructiudelscanvis. Al Capítol 5 s’apliquenelsresultatsdelscapítolsanteriors, combinatsamblinealitzacionsDarbouxianes. Concretament, es considera un sistema quadràtictipusLotka-Volterra i es caracteritzen les selles linealitzables i orbitalmentlinealitzablesmitjançant la troballadelscanvislinealitzants o orbitalmentlinealitzants. En el sisè capítol, s’utilitzal’existènciad’unàlgebra de simetriespuntuals de Lie per donar informació sobre l’existència i localitzaciód’òrbitesperiòdiques. En particular, quanl’àlgebra de simetriespuntuals de Lie d’unaequació diferencial escalar de segónordreautònoma i suau té dimensiómajor o igual a dos, definim les anomenadesfuncionsfonamentals que enspermeten estudiar les òrbitesperiòdiques al pla de fases. En el cas particular d’equacionspolinomials de Liénard, mostrem la no existència de cicles límit en tot el pla de fases. Finalment, al Capítol 7 es relacionen elssistemes reversibles amb el problema del centre aixícomamb el problema de la integrabilitat analítica. Consideremsistemesd’equacionsdiferencialsanalíticsamb centres degenerats i mostrem que poden transformar-se, després d’un reescalat del temps, en un sistema lineal i reversible. El coneixement de integralsprimeresens proporciona un procediment per caracteritzar, en alguns casos, la condició de reversibilitat del centre degenerat. D’altra banda, relacioneml’existència de integralsprimeresanalítiquesamb la reversibilitat orbital analítica en el cas de singularitatsdèbils no degenerades.

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Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG
In this work we exhibit canonical forms for 2D codimension one piecewise smooth vector Fields (PSVF). All possible orientations and codimension one scenarios were covered. Also the intrinsic objects that characterize each one of the canonical forms were presented. Also we present topological distinct canonical forms for a larger class for symmetric PSVF where the set of fixed points is contained in the variety os discontinuity. Finally we analyze the simultaneous occurrence of sliding and crossing limit cycle in the case where the piecewise linear vector fields presents a continuum of periodic orbits.
Neste trabalho exibiremos inicialmente as formas canônicas para campos vetoriais suaves por partes (PSVF) no plano. Todas os possíveis cenários de codimensão um são abordados. Também apresentamos formas canônicas topologicamente distintas para uma classe de PSVF com simetria onde o conjunto de pontos fixos está contido na variedade de descontinuidade. Finalmente, analisaremos a ocorrência simultânea de ciclos limite costurantes e deslizantes no caso linear por partes que apresentam um contínuo de órbitas periódicas.

Aquesta tesi consta d'un primer capítol introductori, set capítols amb diferents resultats i una bibliografia. El primer capítol conté la definició i els resultats previs necessaris per abordar la resta de la memòria. El capítol 2 i 3 estan molt relacionats. En el primer es descriu un mètode alternatiu per al còmput de les constants de Poincaré--Liapunov. A diferència de mètodes anteriors, el mètode presentat no requereix el càlcul d'integrals i d'ona de forma explícita les constants de Poincaré--Liapunov. En el tercer capítol es descriu com s'ha implementat aquest nou mètode i els resultats que d'ona per a sistemes quadràtics i sistemes amb termes no lineals cúbics homogenis. El quart capítol es centra en equacions d'Abel i la seva integrabilitat. Es descriu la forma d'una integral primera que sigui algebraica en funcíó de les variables dependents i es donen múltiples exemples d'equacions d'Abel integrables en aquest sentit. En el cinquè capítol també s'aborda el problema de la integrabilitat però per a equacions diferencials en el pla definides per funcions analítiques. Es fa un reescalat de les variables dependents i de la variable independent amb un paràmetre "epsilon" que està elevat a potències senceres (blow-up paramètric) de forma que el sistema resultant sigui analític en "epsilon". Es d'ona un mètode que aprofita que una integral primera, si existeix, ha de ser analítica en el paràmetre a fi de trobar condicions per a l'existència d'aquesta integral primera. D'aquesta manera es defineix el que s'anomenen variables essencials del sistema. Els darrers tres capítols versen sobre les equacions d'Abel i el problema del centre. En general es consideren equacions d'Abel trigonomètriques. En el sisè capítol es donen algunes condicions necessàries i suficients per a que una equació d'Abel trigonomètrica definida per polinomis trigonomètrics de grau fins a 3 tingui un centre. Tots els exemples donats en aquest capítol tenen un centre universal. En el capítol setè es d'ona un exemple d'una equació d'Abel trigonomètrica definida per polinomis trigonomètrics de grau 3 que té un centre que no és universal. D'aquesta manera es resol un problema obert: determinar el grau més petit pel qual un equació d'Abel trigonomètrica amb centre no és de composició. El darrer capítol tracta equacions d'Abel trigonomètriques i polinomials i d'ona un compendi dels darrers resultats coneguts i conjectures sobre el problema del centre en aquestes equacions. També es donen exemples nous d'equacions d'Abel amb centre.
Esta tesis consta de un primer capítulo introductorio, siete capítulos con diferentes resultados y una bibliografía. El primer capítulo contiene la definición y los resultados previos necesarios para abordar el resto de la memoria. Los capítulos 2 y 3 están muy relacionados. En el primero se describe un método alternativo para el cómputo de las constantes de Poincaré--Liapunov. A diferencia de métodos anteriores, el método presentado no requiere el cálculo de integrales y da de forma explícita las constantes de Poincaré--Liapunov. En el tercer capítulo se describe cómo se ha implementado este nuevo método y los resultados que da para sistemas cuadráticos y sistemas con términos no lineales cíbicos homogéneos. El cuarto capítulo se centra en ecuaciones de Abel y su integrabilidad. Se describe la forma de una integral primera que sea algebraica en función de las variables dependientes y se dan múltiples ejemplos de ecuaciones de Abel integrables en este sentido. En el quinto capítulo también se aborda el problema de la integrabilidad pero para ecuaciones diferenciales en el plano definidas por funciones analíticas. Se hace un reescalado de las variables dependientes y de la variable independiente con un parámetro "epsilon" que está elevado a poténcias enteras (blow-up paramétrico) de forma que el sistema resultante sea analítico en "epsilon". Se da un método que aprovecha que una integral primera, si existe, debe ser analítica en el parámetro con el fin de encontrar condiciones para la existéncia de esta integral primera. De esta manera se define lo que se llaman variables esenciales del sistema. Los últimos tres capítulos versan sobre las ecuaciones de Abel y el problema del centro. En general se consideran ecuaciones de Abel trigonométricas. En el sexto capítulo se dan algunas condiciones necesarias y suficientes para que una ecuación de Abel definida por polinomios trigonométricos de grado hasta 3 tenga un centro. Todos los ejemplos dados en este capítulo tienen un centro universal. En la capítulo séptimo se da un ejemplo de una ecuación de Abel definida por polinomios trigonométricos de grado 3 que tiene un centro que no es universal. De esta manera se resuelve un problema abierto: determinar el grado mas pequeño por el que una ecuación de Abel trigonométrica con centro no es de composición. El último capítulo trata ecuaciones de Abel trigonométricas y polinomiales y da un compendio de los últimos resultados conocidos y conjeturas sobre el problema del centro en estas ecuaciones. También se dan ejemplos nuevos de ecuaciones de Abel con centro.
This thesis consists of a first introductory chapter, seven chapters with different results and a bibliography. The first chapter contains the definition and the previous results necessary to address the rest of the memory. Chapters 2 and 3 are closely related. In the first one, an alternative method is described for the computation of the Poincaré--Liapunov constants. Unlike previous methods, the presented method does not require the computation of primitives and gives an explicit expression of the Poincaré--Liapunov constants. The third chapter describes how this new method has been implemented and the results that it gives for quadratic systems and systems with homogeneous, cubic, non-linear terms. The fourth chapter focuses on Abel equations and their integrability. We describe the form of a first integral that is algebraic in function of the dependent variables and give more examples of equations of Abel integrable from this point of view. The fifth chapter also discusses the integrability problem but for differential equations in the plane defined by analytical functions. A rescaling of the dependent and the independent variables with a parameter "epsilon" which is elevated to integer powers (parametrical blow up) so that the resulting system is analytical in "epsilon". A method is given that takes advantage that a first integral, if it exists, it must be analytical in the parameter in order to find conditions for the existence of this first integral. In this way we define what are called essential variables of the system. The last three chapters deal with Abel equations and the center problem. In general, we consider Abel trigonometric equations. In the sixth chapter some necessary and sufficient conditions for an Abel equation defined by trigonometric polynomials of degree up to 3 have a center are given. All the examples given in this chapter have a universal center. In the seventh chapter it is given an example of an Abel equation defined by trigonometric polynomials of degree 3 with a center which is not universal. In this way an open problem is solved: to determine the lowest degree such that a trigonometric Abel equation has a center which is not a composition center. The last chapter deals with trigonometric and polynomial Abel equations and gives a survey of the last known results and conjectures about the center problem for these equations. Besides some new examples of Abel differential equations with a center are given.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
Nesta dissertação estudamos critérios para determinar a existência, a não existência e a unicidade de ciclos limites de campos de vetores planares. Mais especificamente, estudamos equações de Lienard Äx + f(x; _ x) _ x + g(x) = 0; onde f e g satisfazem determinadas hip¶oteses. Em particular estudamos a equa»c~ao de van der Pol Äx + (x2 ¡ 1) _ x + x = 0; a qual é conhecida da teoria dos circuitos elétricos. Provamos a existência e a unicidade de ciclos limites para estas equações. Por fim estudamos a equação de van der Pol com o parâmetro 1 e o fenômeno canard que ocorre ao considerarmos um parâmetro adicional ®: As técnicas utilizadas s~ao as usuais de Análise Assintótica.
In this work we study the existence, the non existence and the uniqueness of limit cycles of planar vector felds. More specifically, we study Lienard equations Äx+f(x; _ x) _ x+g(x) = 0; where f and g satisfy some hypothesis. In particular we study the van der Pol equation Äx + (x2 ¡ 1) _ x + x = 0; which is knew of the circuit theory. We prove the existence and the uniqueness of limit cycles for these equations. In the last part we study the van der Pol equation with the parameter 1 and the canard phenomenon which appears when we consider an additional parameter ®: The techniques employed are the usual in the Asymptotic Analysis.

Orientador: Luci Any Francisco Roberto
Coorientador: Claudio Aguinaldo Buzzi
Banca: Ana Cristina Mereu
Banca: Claudio Gomes Pessoa
Resumo: Neste trabalho temos como objetivo estudar o número e a distribuição de ciclos limites em sistemas diferenciais lineares por partes. Em particular estudamos o número de ciclos limites do sistema diferencial linear por partes planar ˙x = −y − ε φ ( x) , ˙y = x, onde ε 6= 0 é um parâmetro pequeno e φ é uma função periódica linear por partes ímpar de período 4 . Provamos que dado um inteiro arbitário positivo n, o sistema acima possui exatamente n ciclos limites na faixa |x| ≤ 2 (n + 1 ). Consequentemente, existem sistemas diferenciais lineares por partes contendo uma infinidade de ciclos limites no plano real. Inicialmente obtemos uma quota inferior par a o número destes ciclos limites na faixa | x| ≤ 2 (n + 1 ) via Teoria do Averaging . Em seguida , utilizando a Teoria de Campos de Vetores Rodados, verificamos que o sistema acima tem exatamente n ciclos limites na faixa | x| ≤ 2 (n + 1 )
Abstract: The main goal of this work aim to study the number and distribution of limit cycles in piecewise linear differential systems. In particular we consider the planar piecewise linear differential system ˙x = −y − ε φ ( x) , ˙y = x, where ε 6= 0 is a small parameter and φ is an odd piecewise linear periodic function of period 4 . We prove that given an arbitrary positive integer n, the system above has exactly n limit cycles in the strip | x| ≤ 2 (n + 1 ) . Consequently, there are piecewise differential systems containing an infinite number of limit cycles in the real plane. First we get a lower bound on the number of limit cycles in the strip |x| ≤ 2 (n + 1 ) via Averaging Theory. In the following , using the Theory of Rotated Vector Fields, we see that above system has exactly n limit cycles in the strip | x| ≤ 2 (n + 1 )
Mestre

A primeira parte deste trabalho é dedicada ao estudo de sistemas dinâmicos contínuos e discretos bidimensionais com um único ponto de equillíbrio que é do tipo sela hiperbólica. No caso contínuo, obtemos condições sufiientes para que um campo vetorial planar seja topologicamente equivalente à sela linear L(x; y) = (-x; y). No caso em que o campo vetorial é um difeomorfismo local, a injetividade do campo jogará um papel fundamental na obtenção de tal equivalência topológica. Além disto, apresentamos uma descrição das folheações do plano associadas a campos de vetores com uma única singularidade do tipo sela hiperbólica. No âmbito dos sistemas discretos, apresentamos condições para que um difeomorfismo, possuindo uma sela hiperbólica como único ponto fixo, satisfaça as propriedades básicas de um sistema linear com um ponto fixo que é do tipo sela hiperbólica: as quatro separatrizes do ponto fixo se acumulam só no infinito e os iterados dos pontos que não estão nas variedades invariantes deste ponto fixo se acumulam no infinito tanto no passado quanto no futuro. A segunda parte deste texto, se dedica a problemas de injetividade de difeomorfismos locais em \'R POT. n\'. Mais especificamente, obtemos versões fracas da Conjetura Jacobiana Real de Jelonek e de uma Conjetura apresentada por Nollet e Xavier. Ambos problemas estão intimamente ligados à famosa Conjetura Jacobiana, que foi considerada por Smale em 1998 como um dos dezoito problemas matemáticos mais relevantes ainda em aberto
The first part of this work is dedicated to the study of continuous and discrete twodimensional dynamical systems with a unique equilibrium point which is a hyperbolic saddle. In the continuous case, we obtain sufficient conditions for a planar vector field be topologically equivalent to the linear saddle L(x; y) = (-x; y). In the case where the vector field is a local diffeomorphism, the injectivity of the field will play a key role in obtaining such a topological equivalence. Furthermore, we provide a description of foliations of the plane vector fields associated with a unique singularity of hyperbolic saddle type. In the context of discrete systems, we present conditions for a diffeomorphism, possessing a hyperbolic saddle as the single fixed point, to satisfy the basic properties of a linear system with a fixed point of saddle type which is hyperbolic: the four separatrices of the fixed point accumulate only at infinity and iterated the points that are not in invariant manifolds of this fixed point accumulate in infinity in both the past and future. The second part of this text is devoted to problems of injectivity of local diffeomorphisms on \'R POT. n\'. More specifically, we obtain weaker versions of the Jelonek\'s Real Jacobian Conjecture and a Conjecture given by Nollet and Xavier. Both problems are closely linked to the famous Jacobian Conjecture, which was considered by Smale in 1998 as one of eighteen mathematical problems even more important in open

Le but de cette thèse est d’étudier, à l’aide d’outils combinatoires simples, différentes structures géométriques construites à partir de l’action d’un polynôme ou d’une fraction rationnelle. Nous considérerons d’abord la structure de l'ensemble des solutions séparatrices d’un champ de vecteurs polynomial ou rationnel. Nous allons établir plusieurs modèles combinatoires de ces cartes planaires, ainsi qu’une formule fermée énumérant les différentes structures topologiques dans le cas polynomial. Puis nous parlerons de revêtements ramifiés de la sphère que nous modéliserons, via un objet combinatoire nommée carte équilibrée, à partir d’une idée originale de W.Thurston. Ce modèle nous permettra de démontrer (géométriquement) de nombreuses propriétés de ces objets, et d’offrir une nouvelle approche et de nouvelles perspectives au problème d’Hurwitz, qui reste encore aujourd’hui un problème ouvert. Et enfin nous aborderons le sujet de la dynamique holomorphe via les primitives majeures dont l’utilité est de permettre de paramétrer les systèmes dynamiques engendrés par l’itération de polynômes. Cette approche nous permettra de construire une bijection entre les suites de parking et les arbres de Cayley, ainsi que d’établir une formule fermée liée à l’énumération d’un certain type d’arbres relié à la fois aux primitives majeures et aux revêtements ramifiés polynomiaux
The main topic of this thesis is to study, thanks to simple combinatorial tools, various geometric structures coming from the action of a complex polynomial or a rational function on the sphere. The first structure concerns separatrix solutions of polynomial or rational vector fields. We will establish several combinatorial models of these planar maps, as well as a closed formula enumerating the different topological structures that arise in the polynomial settings. Then, we will focus on branched coverings of the sphere. We establish a combinatorial coding of these mappings using the concept of balanced maps, following an original idea of W. Thurston. This combinatorics allows us to prove (geometrically) several properties about branched coverings, and gives us a new approach and perspective to address the still open Hurwitz problem. Finally, we discuss a dynamical problem represented by primitive majors. The utility of these objects is to allow us to parameterize dynamical systems generated by the iterations of polynomials. This approach will enable us to construct a bijection between parking functions and Cayley trees, and to establish a closed formula enumerating a certain type of trees related to both primitive majors and polynomial branched coverings

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
O principal resultado dessa dissertação é o Teorema de Poincaré-Bendixson para campos de vetores planares suaves por partes, que nos diz quais são os tipos de conjuntos limite. Estudaremos também detalhes a respeito dos conceitos de conjuntos minimais e caóticos em campos de vetores planares suaves por partes.
The main result of this work is the Poincaré - Bendixson Theorem for planar piecewise smooth vector fields, which tell us what kind of limit sets arise in this context. We will also study details about the concepts of minimal and chaotic sets in planar piecewise smooth vector fields.

Orientador: Paulo Ricardo da Silva
Banca: Luis Fernando Mello
Banca: João Carlos Ferreira Costa
Resumo: Nesta dissertação estudamos critérios para determinar a existência, a não existência e a unicidade de ciclos limites de campos de vetores planares. Mais especificamente, estudamos equações de Lienard Äx + f(x; _ x) _ x + g(x) = 0; onde f e g satisfazem determinadas hip¶oteses. Em particular estudamos a equa»c~ao de van der Pol Äx + "(x2 ¡ 1) _ x + x = 0; a qual é conhecida da teoria dos circuitos elétricos. Provamos a existência e a unicidade de ciclos limites para estas equações. Por fim estudamos a equação de van der Pol com o parâmetro" " 1 e o fenômeno canard que ocorre ao considerarmos um parâmetro adicional ®: As técnicas utilizadas s~ao as usuais de Análise Assintótica.
Abstract: In this work we study the existence, the non existence and the uniqueness of limit cycles of planar vector felds. More specifically, we study Lienard equations Äx+f(x; _ x) _ x+g(x) = 0; where f and g satisfy some hypothesis. In particular we study the van der Pol equation Äx + "(x2 ¡ 1) _ x + x = 0; which is knew of the circuit theory. We prove the existence and the uniqueness of limit cycles for these equations. In the last part we study the van der Pol equation with the parameter " " 1 and the canard phenomenon which appears when we consider an additional parameter ®: The techniques employed are the usual in the Asymptotic Analysis.
Mestre

Due to the ever-increasing worldwide consumption of memory for digital information, new technologies for higher capacity and faster data storage systems have been the focus of research and development. A step towards achieving higher data storage densities or magnetic recording media is the concept of bit patterned media, where the magnetic recording layer is divided up into magnetically isolated bit units. This approach is one of the most promising technologies for increasing data storage densities and could be implemented by nanostructuring the wafer. Therefore, the fabrication of the appropriate nanostructures on a small scale and then be able to manufacture these structures on an industrial scale is one of the problems where science and industry are working on a solution. In addition, the answer to the open question about the influence that patterning on the nano length scale has on the magnetic properties is of great interest. The main goal of this thesis is to answer the open question, which magnetic properties can be tailored by a modification of the surface texture on the nanometre length scale. For this purpose the following properties: anisotropy, remanence, coercivity, switching field distribution, saturation magnetisation, Gilbert damping, and inhomogeneous linebroadening were compared between planar two dimensional thin ferromagnetic films and three dimensional magnetic structures. In addition, the influences of the tailored morphology on the intergranular or the exchange coupling between the structures, which is called interdot exchange coupling, was investigated. For the ferromagnetic thin films, the focus of the investigations was on the granular CoCrPt:SiO2 and [Co/Pd] layer, which currently are the state-of-the-art material for magnetic data storage media. These materials are characterised by their high coercivity and high perpendicular anisotropy, which has a low spatial distribution in the preferred direction of magnetisation. In this work the pre-structured GaSb(001) substrate with self-assembled periodic nanocone structures at the surface are used. The preparation by ion beam erosion of these structures is simple, fast, and highly reproducible and therefore this method is particularly beneficial for fundamental research. To compare the 2D thin films with the 3D magnetic structures, besides the pre-structured specimen, planar samples were also fabricated. The first sample series prepared was coated by Py. Due to the fact that the magnetic properties of this material are well-known, it was also possible to do some OOMMF simulations in addition to the VNA-FMR and MOKE measurements. Afterwards two planar samples with CoCrPt and CoCrPt:SiO2 were prepared. The planar CoCrPt:SiO2 samples were Co+ ion implanted to study the influence of such irradiation on the intergranular and interdot exchange coupling, switching field distribution, and in particular on the spin dynamics. Moreover, both samples were measured by TRMOKE in order to obtain information about the spin dynamics. Subsequently, the perpendicular storage media materials CoCrPt:SiO2 and [Co/Pd] were deposited on a prestructured GaSb(001) nanocone substrate surface. These sample series were measured by MOKE, SQUID, and vector-VSM. The measurements demonstrate the influence of the periodicity and height of the nanocones on the intergranular and interdot exchange coupling. They also show the reorientation of the magnetisation with respect to the curvature of the substrate template and furthermore, the morphology-induced influences on the magnetic domains. From the comparison between the results for the planar and the pre-structured samples, a decrease of the interdot exchange coupling was observed, which scales together with the periodicity of the nanocone pattern. In addition, it was shown that for all samples with thin magnetic films on nanocones,the magnetisation aligns along the curvature of the underlying nanocone structure. For Py on nanocones, planar granular CoCrPt:SiO2, and planar granular CoCrPt, measurements by VNA-FMR and TRMOKE could be carried out, which yielded information about the spin dynamics. The results obtained for both of the planar sample are comparable to values from the literature for the Gilbert damping. The results for the Py samples showed that the commonly used 2D model resonance condition is, in case of a 3D magnetic structure, no longer valid due to the alignment of the magnetisation along the underlying substrate structure and therefore an new model has to be derived
Aufgrund des weltweiten, immer weiter steigenden Bedarfs an Speicherplatz von digitalen Information, sind neue Technologien für größere und schnellere Speichermedien im Fokus von Forschung und Entwicklung. Ein Schritt hin zu einer höheren Speicherdichte in der magnetischen Datenspeicherung ist dabei das sogenannte Konzept der ”Bit patterned media”, das definierte Informationseinheiten auf regelmäßig angeordneten Nanostrukturen beschreibt. Dieser Ansatz ist einer der derzeit vielversprechendsten Optionen die Speicherdichte zu erhöhen. Dabei ist die Herstellung der benötigten Nanostrukturen und deren Skalierung hin zu makroskopischen Dimensionen eines der Probleme an deren Lösung die Wissenschaft und Industrie derzeit arbeitet. Desweiteren ist die Antwort auf die noch offene Frage nach der Beeinflussung der nanoskaligen Strukturen auf die magnetischen Eigenschaften von großem Interesse. Das Hauptziel in dieser Arbeit ist es, einen Beitrag zur Beantwortung der Frage, welche magnetischen Eigenschaften sich durch eine Veränderung der Oberflächenstruktur im Nanometerbereich beeinflussen lassen, zu leisten. Hierzu wurden die folgenden Eigenschaften, wie zum Beispiel die Anisotropie, Remanenz,Koerzitivität, Schaltfeldverteilung, Sättigungsmagnetisierung, Gilbertdämpfung und inhomogene Linienverbreiterung von planaren zweidimensionalen dünnen ferromagnetische Schichten mit denen von dreidimensionalen magnetischen Strukturen verglichen. Zusätzlich wurde der Einfluss der angegpassten Morphologie auf die intergranularen- beziehungsweise auf die zwischen den Strukturen wirkende (interdot) Austauschkopplung untersucht. Der Hauptaugenmerk bei den ferromagnetisch dünnen Schichten lag dabei auf den granularen CoCrPt:SiO2 und [Co/Pd] Filmen, die heutzutage ein Standardmaterial für die magnetischen Speichermedien darstellen. Diese Materialien zeichnen sich durch eine hohe Koerzivität und senkrechte Anisotropie, mit geringer räumlicher Verteilung der Vorzugsrichtung der Magnetisierung, aus. Die hier vorgestellten vorstrukturierten GaSb(001) Substrate mit selbstordnenden periodischen Nanokegeln auf der Oberfläche, sind mittels Ionenstrahlerosion einfach, schnell und sehr gut reproduzierbar herzustellen. Deshalb ist diese Methode besonders für die Grundlagenforschung von Vorteil. Um einen Vergleich zwischen 2D Filmen und 3D Strukturen ziehen zu können, wurden neben den vorstrukturierten Substraten auch planare Proben beschichtet. Eine erste Versuchsreihe wurde mit einem dünnen Py Film präpariert. Da dessen magnetische Eigenschaften wohlbekannt sind, konnten neben den Untersuchungen mit VNA-FMR und MOKE auch einige OOMF Simulationen erstellt werden. Danach wurden zwei Proben mit planarem CoCrPt beziehungsweise CoCrPt:SiO2 untersucht. Bei den planaren CoCrPt:SiO2 Proben wurden außerdem noch Co+ Ionen implantiert, um deren Auswirkungen auf die intergranulare Austauschkopplung, Schaltfeldverteilung und besonders auf die Spindynamik zu bestimmen. Bei beiden Probensystemen konnte zusätzlich die Spindynamik mittels zeitaufgelöstem MOKE gemessen werden. Im Anschluss wurden die beiden senkrechten Speichermedien CoCrPt:SiO2 and [Co/Pd] auf Substraten mit Nanokegeln vorstrukturierten GaSb(001) Oberflächen abgeschieden. Diese Proben wurden mit MFM, MOKE, SQUID und Vektor-VSM vermessen. Aus den Messungen konnnten dann die Einflüsse auf die intergranulare- beziehungsweise interdot Austauschkopplung in Abhängigkeit von der Periodizität und Höhe der Nanokegel bestimmt werden, sowie die Umorientierung der Magnetisierung bezüglich der Substratkrümmung und den Morphologie induzierten Einfluss auf die magnetischen Domänen. Anhand der Vergleiche zwischen den Messungen der planaren und den vorstrukturierten Proben konnte eine Verringerung der Austauschkopplung zwischen den Strukturen gezeigt werden, die mit der Nanokegelstrukturperiodizität skaliert. Außerdem wurde in allen dünnen magnetischen Filmen auf Nanokegeln gezeigt, dass die Magnetisierung sich in Abhängigkeit der darunterliegenden Struktur ausrichtet. Bei den Py auf Nanokegeln, den planaren CoCrPt und dem planaren CoCrPt:SiO2 Proben konnten außerdem mit VNA-FMR und TRMOKE Informationen bezüglich der Spindynamik gemessen werden. Die erzielten Ergebnisse, der beiden planaren Proben, sind vergleichbar mit denen, aus der Literatur bekannten Werten, für die Gilbertdämpfung. Darüber hinaus wurde durch die Messungen an den Py Proben gezeigt, dass die Theorie, des bisher genutzten 2D Modells, nicht mehr gültig ist, da sich die Magnetisierung entlang der Substratstruktur ausrichtet, und deshalb ein neues Model aufgestellt werden muss

Due to the ever-increasing worldwide consumption of memory for digital information, new technologies for higher capacity and faster data storage systems have been the focus of research and development. A step towards achieving higher data storage densities or magnetic recording media is the concept of bit patterned media, where the magnetic recording layer is divided up into magnetically isolated bit units. This approach is one of the most promising technologies for increasing data storage densities and could be implemented by nanostructuring the wafer. Therefore, the fabrication of the appropriate nanostructures on a small scale and then be able to manufacture these structures on an industrial scale is one of the problems where science and industry are working on a solution. In addition, the answer to the open question about the influence that patterning on the nano length scale has on the magnetic properties is of great interest. The main goal of this thesis is to answer the open question, which magnetic properties can be tailored by a modification of the surface texture on the nanometre length scale. For this purpose the following properties: anisotropy, remanence, coercivity, switching field distribution, saturation magnetisation, Gilbert damping, and inhomogeneous linebroadening were compared between planar two dimensional thin ferromagnetic films and three dimensional magnetic structures. In addition, the influences of the tailored morphology on the intergranular or the exchange coupling between the structures, which is called interdot exchange coupling, was investigated. For the ferromagnetic thin films, the focus of the investigations was on the granular CoCrPt:SiO2 and [Co/Pd] layer, which currently are the state-of-the-art material for magnetic data storage media. These materials are characterised by their high coercivity and high perpendicular anisotropy, which has a low spatial distribution in the preferred direction of magnetisation. In this work the pre-structured GaSb(001) substrate with self-assembled periodic nanocone structures at the surface are used. The preparation by ion beam erosion of these structures is simple, fast, and highly reproducible and therefore this method is particularly beneficial for fundamental research. To compare the 2D thin films with the 3D magnetic structures, besides the pre-structured specimen, planar samples were also fabricated. The first sample series prepared was coated by Py. Due to the fact that the magnetic properties of this material are well-known, it was also possible to do some OOMMF simulations in addition to the VNA-FMR and MOKE measurements. Afterwards two planar samples with CoCrPt and CoCrPt:SiO2 were prepared. The planar CoCrPt:SiO2 samples were Co+ ion implanted to study the influence of such irradiation on the intergranular and interdot exchange coupling, switching field distribution, and in particular on the spin dynamics. Moreover, both samples were measured by TRMOKE in order to obtain information about the spin dynamics. Subsequently, the perpendicular storage media materials CoCrPt:SiO2 and [Co/Pd] were deposited on a prestructured GaSb(001) nanocone substrate surface. These sample series were measured by MOKE, SQUID, and vector-VSM. The measurements demonstrate the influence of the periodicity and height of the nanocones on the intergranular and interdot exchange coupling. They also show the reorientation of the magnetisation with respect to the curvature of the substrate template and furthermore, the morphology-induced influences on the magnetic domains. From the comparison between the results for the planar and the pre-structured samples, a decrease of the interdot exchange coupling was observed, which scales together with the periodicity of the nanocone pattern. In addition, it was shown that for all samples with thin magnetic films on nanocones,the magnetisation aligns along the curvature of the underlying nanocone structure. For Py on nanocones, planar granular CoCrPt:SiO2, and planar granular CoCrPt, measurements by VNA-FMR and TRMOKE could be carried out, which yielded information about the spin dynamics. The results obtained for both of the planar sample are comparable to values from the literature for the Gilbert damping. The results for the Py samples showed that the commonly used 2D model resonance condition is, in case of a 3D magnetic structure, no longer valid due to the alignment of the magnetisation along the underlying substrate structure and therefore an new model has to be derived.
Aufgrund des weltweiten, immer weiter steigenden Bedarfs an Speicherplatz von digitalen Information, sind neue Technologien für größere und schnellere Speichermedien im Fokus von Forschung und Entwicklung. Ein Schritt hin zu einer höheren Speicherdichte in der magnetischen Datenspeicherung ist dabei das sogenannte Konzept der ”Bit patterned media”, das definierte Informationseinheiten auf regelmäßig angeordneten Nanostrukturen beschreibt. Dieser Ansatz ist einer der derzeit vielversprechendsten Optionen die Speicherdichte zu erhöhen. Dabei ist die Herstellung der benötigten Nanostrukturen und deren Skalierung hin zu makroskopischen Dimensionen eines der Probleme an deren Lösung die Wissenschaft und Industrie derzeit arbeitet. Desweiteren ist die Antwort auf die noch offene Frage nach der Beeinflussung der nanoskaligen Strukturen auf die magnetischen Eigenschaften von großem Interesse. Das Hauptziel in dieser Arbeit ist es, einen Beitrag zur Beantwortung der Frage, welche magnetischen Eigenschaften sich durch eine Veränderung der Oberflächenstruktur im Nanometerbereich beeinflussen lassen, zu leisten. Hierzu wurden die folgenden Eigenschaften, wie zum Beispiel die Anisotropie, Remanenz,Koerzitivität, Schaltfeldverteilung, Sättigungsmagnetisierung, Gilbertdämpfung und inhomogene Linienverbreiterung von planaren zweidimensionalen dünnen ferromagnetische Schichten mit denen von dreidimensionalen magnetischen Strukturen verglichen. Zusätzlich wurde der Einfluss der angegpassten Morphologie auf die intergranularen- beziehungsweise auf die zwischen den Strukturen wirkende (interdot) Austauschkopplung untersucht. Der Hauptaugenmerk bei den ferromagnetisch dünnen Schichten lag dabei auf den granularen CoCrPt:SiO2 und [Co/Pd] Filmen, die heutzutage ein Standardmaterial für die magnetischen Speichermedien darstellen. Diese Materialien zeichnen sich durch eine hohe Koerzivität und senkrechte Anisotropie, mit geringer räumlicher Verteilung der Vorzugsrichtung der Magnetisierung, aus. Die hier vorgestellten vorstrukturierten GaSb(001) Substrate mit selbstordnenden periodischen Nanokegeln auf der Oberfläche, sind mittels Ionenstrahlerosion einfach, schnell und sehr gut reproduzierbar herzustellen. Deshalb ist diese Methode besonders für die Grundlagenforschung von Vorteil. Um einen Vergleich zwischen 2D Filmen und 3D Strukturen ziehen zu können, wurden neben den vorstrukturierten Substraten auch planare Proben beschichtet. Eine erste Versuchsreihe wurde mit einem dünnen Py Film präpariert. Da dessen magnetische Eigenschaften wohlbekannt sind, konnten neben den Untersuchungen mit VNA-FMR und MOKE auch einige OOMF Simulationen erstellt werden. Danach wurden zwei Proben mit planarem CoCrPt beziehungsweise CoCrPt:SiO2 untersucht. Bei den planaren CoCrPt:SiO2 Proben wurden außerdem noch Co+ Ionen implantiert, um deren Auswirkungen auf die intergranulare Austauschkopplung, Schaltfeldverteilung und besonders auf die Spindynamik zu bestimmen. Bei beiden Probensystemen konnte zusätzlich die Spindynamik mittels zeitaufgelöstem MOKE gemessen werden. Im Anschluss wurden die beiden senkrechten Speichermedien CoCrPt:SiO2 and [Co/Pd] auf Substraten mit Nanokegeln vorstrukturierten GaSb(001) Oberflächen abgeschieden. Diese Proben wurden mit MFM, MOKE, SQUID und Vektor-VSM vermessen. Aus den Messungen konnnten dann die Einflüsse auf die intergranulare- beziehungsweise interdot Austauschkopplung in Abhängigkeit von der Periodizität und Höhe der Nanokegel bestimmt werden, sowie die Umorientierung der Magnetisierung bezüglich der Substratkrümmung und den Morphologie induzierten Einfluss auf die magnetischen Domänen. Anhand der Vergleiche zwischen den Messungen der planaren und den vorstrukturierten Proben konnte eine Verringerung der Austauschkopplung zwischen den Strukturen gezeigt werden, die mit der Nanokegelstrukturperiodizität skaliert. Außerdem wurde in allen dünnen magnetischen Filmen auf Nanokegeln gezeigt, dass die Magnetisierung sich in Abhängigkeit der darunterliegenden Struktur ausrichtet. Bei den Py auf Nanokegeln, den planaren CoCrPt und dem planaren CoCrPt:SiO2 Proben konnten außerdem mit VNA-FMR und TRMOKE Informationen bezüglich der Spindynamik gemessen werden. Die erzielten Ergebnisse, der beiden planaren Proben, sind vergleichbar mit denen, aus der Literatur bekannten Werten, für die Gilbertdämpfung. Darüber hinaus wurde durch die Messungen an den Py Proben gezeigt, dass die Theorie, des bisher genutzten 2D Modells, nicht mehr gültig ist, da sich die Magnetisierung entlang der Substratstruktur ausrichtet, und deshalb ein neues Model aufgestellt werden muss.