We examine an early model for the interaction of HIV with CD4+ T cells in vivo and define possible parameters and effects of said parameters on the model. We then examine a newer, more simplified model for the interaction of HIV with CD4+ T cells that also considers four populations: uninfected T cells, latently infected T cells, actively infected T cells, and free virus. The stability of both the disease free steady state and the endemically infected steady state are examined utilizing standard methods and the Routh-Hurwitz criteria. We show that if N, the number of infectious virions produced per actively infected T cell, is less than a critical value, , then the uninfected state is the only steady state in the non negative orthant, and this state is stable. We establish an expression for . If , then the uninfected steady state is unstable, and the endemically infected state can be stable or unstable, depending on the value of the parameters utilized.

No description available.

Αντικείμενο της παρούσης διπλωματικής εργασίας αποτελεί το Φωτοβαρυτικό Πρόβλημα των Τριών Σωμάτων, ένα άλυτο πρόβλημα που απασχόλησε και εξακολουθεί να απασχολεί τον τομέα των Εφαρμοσμένων Μαθηματικών και της Κλασικής Αστροφυσικής τουλάχιστον τους τελευταίους δύο αιώνες. Διεξάγεται μελέτη των σημείων ισορροπίας του συστήματος και προσδιορίζονται οικογένειες περιοδικών λύσεων, οι οποίες στην περίπτωσή μας διακρίνονται σε δύο κατηγορίες που χαρακτηρίζονται από το μέγεθος της περιόδου τους. / The topic of this thesis deals with the Restricted Photogravitational Three Body Problem, which is an unsolved problem in Astrophysics and Celestial Mechanics. Research is carried out concerning the equilibrium points, around which families of periodic solutions can be identified.

Dans cette thèse on s'intéresse a l'aspect spatial dans la modélisation en épidémiologie, ainsi que des conditions menant a la stabilité des systèmes que nous présentons, en épidémiologie, a partir des modèles classiques de Ross et Mckendrick. Dans un premier temps, nous examinons les effets de l'indice de la différence normalisée de végétation (NDVI) dans un modèle de contamination du paludisme a Bankoumana, une région du Mali. A partir du système obtenu, nous trouvons le taux de reproduction de base. Deux points d'équilibre sont déduits dont, un point d'équilibre sans maladie et un point d'équilibre endémique. Ce dernier point d'équilibre ainsi que le taux de reproduction de base sont fonction de l'indice de végétation normalisée. Par la suite, nous construisons un modèle ayant des équations a retard, dans lequel est également incorporée le NDVI. Le taux de reproduction de base ainsi que les deux points d'équilibre qui découlent de notre système sont fonction des retards introduits. Nous montrons que la stabilité de nos points d'équilibre est, non seulement fonction du taux de reproduction de base, mais elle est aussi étroitement liée aux retards introduits. Dans une autre optique, nous fractionnons la région d'étude en zones dans lesquelles nous émettons l'hypothèse que le taux de contagion induite par les individus d'une zone sur les individus de la même zone, ainsi que celui des individus d'une zone sur ceux d'une autre zone, peut être différents. Nous obtenons un système qui nous permet de déterminer les points d'équilibre ainsi que les conditions qui nous permettent d'obtenir la stabilité au sens de Lyapunov. Puis, nous perturbons le système précédent au niveau de son unique point d'équilibre endémique, en introduisant un bruit additif. Par suite, les conditions permettant la stabilité au sens de Lyapunov, sur le nouveau système obtenu, sont également déduites . Dans un cadre similaire, nous élaborons un modèle multi-groupes, dans lequel nous introduisons des coordonnées spatiales. Les groupes sont formés selon une proximité dépendant du rayon d'un cercle, de manière aléatoire. Ici, le taux de contagion est supposé uniforme dans les groupes. Après avoir déterminé les points d'équilibre ainsi que le taux de reproduction de base,nous trouvons les conditions qui favorisent la stabilité au sens de Lyapunov dans le cadre général. A l'ordre 1, c'est-a-dire, lorsqu'on suppose que nous n'avons qu'un groupe, les conditions de stabilité sont obtenus par le critère de Routh-Hurwitz. / In this thesis, our interest is on the aspect in space of the establishment of a spatial model in epidemiology and the conditions leading to the stability of the systems that we present, in epidemiology, from the classical models by Ross and Mckendrick. Firstly, we intend to examine the eects of the Normalized DiFerence Vegetation Index(NDVI) in a model of contamination of malaria in Bankoumana, a region in Mali. From the system obtained, we willnd the basic reproduction rate. Then we deduce two point of equilibrium, among which one point of equilibrium without the disease and another one with an endemic point. The latter with the basic reproduction rate vary according to the indices of normalized vegetation. Then, we will build a model having equations delay, containing the NDVI. The rate of basic reproduction and the two points of equilibrium that come from our system depend upon the delay introduced. We will show that the stability of our points of equilibrium is not only dependent upon the basic reproduction rate, but also closely related to the delays introduced. In another way, we will divide the region of study in areas where we will set hypotheses that the rate of contamination brought about by individuals in an area of study on the others, can be dierent. It will permit us to obtain a system in which we will determine the points of equilibrium and the conditions that will lead us to obtain the stability according to Lyapunov. Then, we will disturb the previous system at the level of its unique endemic point of equilibrium, with the introduction of an additional noise. The conditions leading to stability according to Lyapunov, on the new system obtained, are generally deduced here. In a similar framework, we will elaborate a multigroups model, in which we will introduce spatial coordinates. The groups are formed according to a closeness depending to a radius of a circle at random. Here, the rate of contamination is supposed to be uniform in the groups. After having determined the point of equilibrium and the rate of basic reproduction, we will nd the conditions facilitating stability in as by Lyapunov in a global framework. In the order1, it means that supposing that we have only one group, the conditions of stability are obtained according to the Routh-Hurvitz criteria.

Submitted by Tamires dos Santos de Moura null (tamiresmoura@hotmail.com) on 2016-08-26T16:55:16Z No. of bitstreams: 1 dissertacao.pdf: 3467990 bytes, checksum: 5dda701bb1d00619df06b7d38eef0bc1 (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-08-29T20:39:29Z (GMT) No. of bitstreams: 1 moura_ts_me_guara.pdf: 3467990 bytes, checksum: 5dda701bb1d00619df06b7d38eef0bc1 (MD5) / Made available in DSpace on 2016-08-29T20:39:29Z (GMT). No. of bitstreams: 1 moura_ts_me_guara.pdf: 3467990 bytes, checksum: 5dda701bb1d00619df06b7d38eef0bc1 (MD5) Previous issue date: 2016-07-14 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Tendo em vista que asteroides são objetos remanescentes dos primórdios do Sistema Solar, estamos interessados na composição deles. Existem missões que estão sendo analisadas com a finalidade de enviar sondas em direção a asteroides do grupo Near Earth Asteroids (NEAs), que representa uma das mais peculiares classes de objetos no Sistema Solar visto que suas órbitas podem se aproximar ou até mesmo cruzar a terrestre. Esse grupo é considerado representativo da população de asteroides, uma vez que podem fornecer informações sobre a mistura química a partir da qual os planetas teriam se formado a bilhões de anos atrás, possibilitando a compreensão da origem e evolução do Sistema Solar e quem sabe até a origem da vida na Terra. Dessa forma, um estudo detalhado a fim de compreender a superfície, a composição e a estrutura interna de um NEA será um grande passo para a Ciência. Nessa pesquisa, inicialmente reproduzimos os dados do potencial gravitacional pelo método dos poliedros para o asteroide 2063 Bacchus, um NEA, a fim de validar os resultados encontrados em Moura (2014). O método dos poliedros fornece uma precisão muito boa da forma irregular do corpo. Por meio de estudo dos modelos de potenciais gravitacionais para corpos não esféricos e implementação de rotinas computacionais foi realizada uma breve análise em relação ao formato do asteroide 2063 Bacchus, bem como das suas superfícies equipotenciais e curvas de velocidade zero. Os objetivos dessa dissertação são realizar um estudo detalhado a respeito dos pontos de equilíbrio no campo gravitacional de 2063 Bacchus, bem como da estabilidade desses pontos levando em consideração os autovalores da equação característica. Além disso, alteramos os valores do período de rotação e da densidade desse objeto a fim de verificar como a localização e a estabilidade dos pontos de equilíbrio alteram quando um parâmetro é mudado. A motivação principal é realizar um estudo o mais realista possível e, dessa forma, observar também como os pontos de equilíbrio se comportam quando introduzimos o efeito da força de pressão de radiação solar que, nesse caso, passam a ser chamados de pontos equivalentes. O trabalho possibilita ampliação do conhecimento não somente para o caso de asteroides, mas também para outros corpos não esféricos como cometas, contribuindo para o desenvolvimento de estudos direcionados a origem e evolução do Sistema Solar. / Given that asteroids are remnant objects of the Solar system beginnings, we are interested in their composition. There are missions that are being analyzed with the purpose of sending probes toward asteroids from the group Near Earth Asteroids (NEAs), which is one of the most peculiar classes of objects in the solar system because their orbits can approach or even cross the Earth’s orbit. This group is considered representative of the population of asteroids, since they can provide information about the chemical mixture from which the planets would have been formed billions of years ago, enabling the understanding of the origin and evolution of the Solar System and maybe even on the origin of life on Earth. Thus a detailed study in order to understand the surface, the composition and internal structure of a NEA will be a big step for Science. In this research, initially we reproduce the data of the gravitational potential by the method of polyhedra for asteroid 2063 Bacchus, a NEA, in order to validate the results found Moura (2014). The method of polyhedra provides a very good accuracy of the irregular shape of the body. Through study of gravitational potential designs for non-spherical bodies and computational routines implementing a brief analysis was performed with respect to the asteroid shape of 2063 Bacchus, as well as its equipotential surfaces and zero-velocity curves. The objectives of this work are to conduct a detailed study on the equilibrium points in the gravitational field of 2063 Bacchus, and the stability of these points taking into account the eigenvalues of the characteristic equation. In addition, we varied the values of the rotation period and density of the object in order to see how the location and stability of equilibrium points changed when a parameter is altered. The main motivation is to achieve a more realistic study and thus, also observe how the equilibrium points behave when we introduce the effect of solar radiation pressure force. The new points are called equivalent points. The work enables expansion of the knowledge, not only in the case of asteroids, but also to other non-spherical bodies like comets, contributing to the development of studies addressing the origin and evolution of the solar system.

The theory of deterministic chaos has generated a lot of interest and continues to be one of the much-focused research areas in the field of dynamics today. This is due to its prevalence in essential parts of human lives such as electrical circuits, chemical reactions, the flow of blood through the human system, the weather, etc. This thesis presents a study of the Lorenz equations, a famous example of chaotic systems. In particular, it presents the analysis of the Lorenz equations from stability to chaos and various bifurcation scenarios with numerical and graphical interpretations. It studies concepts of non-linear dynamical systems such as equilibrium points, stability, linearization, bifurcation, Lyapunov function, etc. Finally, it discusses how the Lorenz equations serve as a model for the waterwheel (in detail), and the convection roll for fluid.

Submitted by Franciele Moreira (francielemoreyra@gmail.com) on 2018-09-21T12:14:18Z No. of bitstreams: 2 Dissertação - Lenison Alves de Queiroz - 2018.pdf: 2458759 bytes, checksum: 2c5c2eaaeddd81877e21434dae197d8e (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-09-24T11:11:13Z (GMT) No. of bitstreams: 2 Dissertação - Lenison Alves de Queiroz - 2018.pdf: 2458759 bytes, checksum: 2c5c2eaaeddd81877e21434dae197d8e (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-09-24T11:11:13Z (GMT). No. of bitstreams: 2 Dissertação - Lenison Alves de Queiroz - 2018.pdf: 2458759 bytes, checksum: 2c5c2eaaeddd81877e21434dae197d8e (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-08-20 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The present work deals with mathematical themes called Conley’s theory, differential inclu- sions and Morse theory inserted in this variant is the topological invariant for the region of discontinuity, the Conley index of discontinuous vector fields, where the discontinuities are concentrated on a surface. With this invariant it is possible to predict bifurcation results, as well as results of regularization of the discontinuous field. In Conley’s Theory, one doesn’t investigate only a single invariant set in a system; on the contrary, it is a decomposition of an invariant set into several “smaller” invariant subsets along with the orbits that connect these subsets. The methodology adopted for the research was based on the deductive analy- sis, a method that allowed the determination of the Conley index using tools of differential inclusions, index-pair and Morse theory to arrive at the determination of the homological in- dex. / O presente trabalho trata de temas da matemática denominados a teoria de Conley, inclusões diferenciais e teoria de Morse inserido nesta variante encontra-se o invariante topológico pa- ra a região de descontinuidade, o índice de Conley de campos de vetores descontínuos, onde as descontinuidades estão concentradas numa superfície. Com este invariante é possível pre- ver resultados de bifurcação, bem como resultados de regularização de campos descontínuos. Na Teoria de Conley, não se investiga somente um único conjunto invariante em um siste- ma, pelo contrário, trata-se de uma decomposição de um conjunto invariante em vários sub- conjuntos invariantes "menores" juntamente com as órbitas que conectam estes subconjuntos. A metodologia adotada para a pesquisa se fundamentou na análise dedutiva, método que per- mitiu determinar o índice de Conley utilizando ferramentas de inclusões diferenciais, par-ín- dice e a teoria de Morse para se chegar a determinação do índice homológico.

Στην παρούσα διατριβή ασχολούμαστε με την μελέτη περιοδικών και ασυμπτωτικών λύσεων στο περιορισμένο πρόβλημα των τεσσάρων σωμάτων. Πιο συγκεκριμένα: Στο κεφάλαιο 1 περιγράφουμε το πρόβλημα των τριών και των τεσσάρων σωμάτων, κάνοντας μια ιστορική αναδρομή και παραθέτουμε τις αρχικές εξισώσεις της κίνησης. Στο κεφάλαιο 2 μελετάμε αριθμητικά το περιορισμένο πρόβλημα των τεσσάρων σωμάτων, στην Lagrangian διαμόρφωση. Υπολογίζουμε τα σημεία ισορροπίας, καθώς και τις επιτρεπτές περιοχές κίνησης του τέταρτου σώματος. Στο κεφάλαιο 3 μελετάμε την ευστάθεια των σημείων ισορροπίας. Επίσης υπολογίζουμε και παρουσιάζουμε τις περιοχές έλξης, για το δυναμικό σύστημα των τεσσάρων σωμάτων. Στο κεφάλαιο 4 μελετάμε οικογένειες απλών συμμετρικών και μη συμμετρικών περιοδικών τροχιών του περιορισμένου προβλήματος των τεσσάρων σωμάτων. Υπολογίζουμε για κάθε περίπτωση τιμών των μαζών, σειρές κρίσιμων περιοδικών τροχιών κάθε οικογένειας ξεχωριστά. Τέλος στο κεφάλαιο 5 μελετάμε αριθμητικά οικογένειες απλών ασύμμετρων περιοδικών τροχιών στο περιορισμένο πρόβλημα των τεσσάρων σωμάτων, έχοντας θέσει ως πρωτεύοντα σώματα τους ΄Ηλιο - Δία και έναν Τρωικό Αστεροειδή και θεωρώντας ως τέταρτο αμελητέας μάζας σώμα ένα διαστημόπλοιο. Τα πρωτεύοντα σώματα υπακούουν στην ευσταθή Lagrangian τριγωνική διαμόρφωση. Μελετήσαμε επίσης αναλυτικά και αριθμητικά τις λύσεις στην περιοχή των ευσταθών σημείων ισορροπίας του συστήματος, βρήκαmε οικογένειες περιοδικών λύσεων και μελετήσαμε την γραμμική ευστάθεια τους. Τα αποτελέσματα των κεφαλαίων 2,3,4 και 5 έχουν δημοσιευτεί σε τρία διεθνή περιοδικά και ένα κομμάτι του κεφαλαίου 5 παρουσιάστηκε σε διεθνές συνέδριο (με συγγραφείς τους Μπαλταγιάννη Α. και Παπαδάκη Κ.). Πιο συγκεκριμένα η μελέτη των κεφαλαίων 2 και 3 έχει δημοσιευτεί στο περιοδικό “International Journal of Bifurcation and Chaos, 21, 2011, pp. 2179-2193” με τον τίτλο: “Equilibrium Points and their stability in the restricted four-body problem”. Τα αποτελέσματα του κεφαλαίου 4 δημοσιεύτηκαν mε τον τίτλο: “Families of periodic orbits in the restricted four-body problem” στο περιοδικό “Astrophysics and Space Science, 336, 2011, pp. 357-367”. Επίσης το κεφάλαιο 5 υπό τον τίτλο “Periodic solutions in the Sun - Jupiter - Trojan Asteroid - Spacecraft system”, δημοσιεύτηκε στο περιοδικό ”Planetary and Space Science, 75, 2013, pp. 148-157”. Το διεθνές συνέδριο στο οποίο παρουσιάστηκε τμήμα του κεφαλαίου 5 ήταν το : “10th Hellenic Astronomical Conference, Proceedings of the conference held at Ioannina, Greece, 5-8 September 2011, pp. 23-24” και η εργασία είχε τίτλο: “Families of periodic orbits in the Sun - Jupiter - Trojan Asteroid system”. Η παρούσα διατριβή εκπονήθηκε με την οικονομική υποστήριξη του ερευνητικού προγράμματος του Πανεπιστημίου Πατρών: Κ. Καραθεοδωρή. / In this thesis we are concerned with the periodic and asymptotic solutions of the restricted four - body problem. In chapter 1 we describe the three - body and four - body problem, starting with historical information. We also present the needed equations of motion and integrals of the problem. In chapter 2 we study numerically the problem of four - bodies, according to the Lagrangian equilateral triangle configuration. We find the equilibrium points and the allowed regions of motion. In chapter 3 we study the stability of the relative equibrium solutions. We also illustrate the regions of the basins of attraction for the equilibrium points of the present dynamical model. In chapter 4 we present families of simple symmetric and non-symmetric periodic orbits in the restricted four-body problem. Series of critical periodic orbits of each family and in any case of the mass parameters are also calculated. In chapter 5 we study, numerically, families of simple non-symmetric periodic orbits of the restricted four-body problem, where we consider the three primary bodies as Sun, Jupiter and a Trojan Asteroid and as a massless fourth body, a spacecraft. The primary bodies are set in the stable Lagrangian equilateral triangle configuration. We also study analytically the solutions in the neighborhood of the stable equilibrium points and the linear stability of each periodic solution. The results of the chapters 2,3,4 and 5 have been published in three journals and a part of chapter 5 has been presented in an international conference. Chapters 2 and 3 have been published in “International Journal of Bifurcation and Chaos, 21, 2011, pp. 2179-2193” under the title of “Equilibrium Points and their stability in the restricted four-body problem”. Chapter 4 has been titled “Families of periodic orbits in the restricted four- body problem” and published in “Astrophysics and Space Science, 336, 2011, pp. 357-367”. Chapter 5 has been titled “Periodic solutions in the Sun - Jupiter - Trojan Asteroid - Spacecraft system,” and published in “Planetary and Space Science, 75, 2013, pp. 148-157”. The conference was the “10th Hellenic Astronomical Conference, Proceedings of the conference held at Ioannina, Greece, 5-8 September 2011, pp. 23-24” and part of the chapter 5 was presented under the title of “Families of periodic orbits in the Sun - Jupiter - Trojan Asteroid system”. This thesis was compiled while the author was in receipt of “K.Karatheodory” research grant.

Neste trabalho estudamos dinâmica Hamiltoniana em $\\mathbb{R}^4$ restrita a níveis de energia próximos a níveis críticos. Mais precisamente, consideramos uma função Hamiltoniana $H: \\mathbb{R}^4 \\to \\mathbb{R}$ que possui um ponto de equilíbrio do tipo sela-centro $p_c \\in H^{-1}(0)$ e assumimos que $p_c$ pertence a um conjunto singular estritamente convexo $S_0 \\subset H^{-1}(0)$. Então, mostramos que os níveis de energia $H^{-1}(E)$, com $E>0$ suficientemente pequeno, contêm uma $3$-bola fechada $S_E$ próxima a $S_0$ que admite um sistema de seções transversais $F_E$, chamado folheação $2-3$. $F_E$ é uma folheação singular de $S_E$ com conjunto singular formado por duas órbitas periódicas $P_{2,E}\\subset \\partial S_E$ e $P_{3,E}\\subset S_E\\setminus \\partial S_E$. A órbita $P_{2,E}$ é hiperbólica dentro do nível de energia $H^{-1}(E)$, pertence à variedade central do sela-centro $p_c$, tem índice de Conley-Zehnder $2$ e é o limite assintótico de dois planos rígidos de $F_E$ que, unidos com $P_{2,E}$, constituem a $2$-esfera $\\partial S_E$. A órbita $P_{3,E}$ tem índice de Conley-Zehnder $3$ e é o limite assintótico de uma família a um parâmetro de planos de $F_E$ contida em $S_E\\setminus \\partial S_E$. Um cilindro rígido conectando as órbitas $P_{3,E}$ e $P_{2,E}$ completa a folheação $F_E$. Uma vez que $F_E$ é um sistema de seções transversais, todas as suas folhas regulares são transversais ao fluxo Hamiltoniano de $H$. Como consequência da existência de uma tal folheação em $S_E$, concluímos que a órbita hiperbólica $P_{2,E}$ admite pelo menos uma órbita homoclínica contida em $S_E \\setminus \\partial S_E$. / In this work we study Hamiltonian dynamics in $\\mathbb R ^4$ restricted to energy levels close to critical levels. More precisely, we consider a Hamiltonian function $H:\\mathbb R ^4 \\to \\mathbb R$ containing a saddle-center equilibrium point $p_c \\in H^ -1 (0)$ and we assume that $p_c$ lies on a strictly convex singular set $S_0 \\subset H^ -1 (0)$. Then we prove that the energy levels $H^ -1 (E)$, with $E>0$ sufficiently small, contain a closed $3$-ball $S_E$ near $S_0$ admitting a system of transverse sections $F_E$, called a $2-3$ foliation. $F_E$ is a singular foliation of $S_E$ and its singular set consists of two periodic orbits $P_{2,E}\\subset \\partial S_E$ and $P_{3,E}\\subset S_E\\setminus \\partial S_E$. The orbit $P_{2,E}$ is hyperbolic inside the energy level $H^ -1 (E)$, lies on the center manifold of the saddle-center $p_c$, has Conley-Zehnder index $2$ and is the asymptotic limit of two rigid planes of $F_E$, which compose the $2$-sphere $S_E$ together with $P_{2,E}$. The orbit $P_{3,E}$ has Conley-Zehnder index $3$ and is the asymptotic limit of a one parameter family of planes of $F_E$ contained in $S_E \\setminus \\partial S_E$. A rigid cylinder connecting the orbits $P_{3,E}$ and $P_{2,E}$ completes the foliation $F_E$. Since $F_E$ is a system of transverse sections, all its regular leaves are transverse to the Hamiltonian flow of $H$. As a consequence of the existence of such foliation in $S_E$, we conclude that the hyperbolic orbit $P_{2,E}$ admits at least one homoclinic orbit contained in $S_E\\setminus \\partial S_E$.

Neste trabalho estudamos dinâmica Hamiltoniana em $\\mathbb{R}^4$ restrita a níveis de energia próximos a níveis críticos. Mais precisamente, consideramos uma função Hamiltoniana $H: \\mathbb{R}^4 \\to \\mathbb{R}$ que possui um ponto de equilíbrio do tipo sela-centro $p_c \\in H^{-1}(0)$ e assumimos que $p_c$ pertence a um conjunto singular estritamente convexo $S_0 \\subset H^{-1}(0)$. Então, mostramos que os níveis de energia $H^{-1}(E)$, com $E>0$ suficientemente pequeno, contêm uma $3$-bola fechada $S_E$ próxima a $S_0$ que admite um sistema de seções transversais $F_E$, chamado folheação $2-3$. $F_E$ é uma folheação singular de $S_E$ com conjunto singular formado por duas órbitas periódicas $P_{2,E}\\subset \\partial S_E$ e $P_{3,E}\\subset S_E\\setminus \\partial S_E$. A órbita $P_{2,E}$ é hiperbólica dentro do nível de energia $H^{-1}(E)$, pertence à variedade central do sela-centro $p_c$, tem índice de Conley-Zehnder $2$ e é o limite assintótico de dois planos rígidos de $F_E$ que, unidos com $P_{2,E}$, constituem a $2$-esfera $\\partial S_E$. A órbita $P_{3,E}$ tem índice de Conley-Zehnder $3$ e é o limite assintótico de uma família a um parâmetro de planos de $F_E$ contida em $S_E\\setminus \\partial S_E$. Um cilindro rígido conectando as órbitas $P_{3,E}$ e $P_{2,E}$ completa a folheação $F_E$. Uma vez que $F_E$ é um sistema de seções transversais, todas as suas folhas regulares são transversais ao fluxo Hamiltoniano de $H$. Como consequência da existência de uma tal folheação em $S_E$, concluímos que a órbita hiperbólica $P_{2,E}$ admite pelo menos uma órbita homoclínica contida em $S_E \\setminus \\partial S_E$. / In this work we study Hamiltonian dynamics in $\\mathbb R ^4$ restricted to energy levels close to critical levels. More precisely, we consider a Hamiltonian function $H:\\mathbb R ^4 \\to \\mathbb R$ containing a saddle-center equilibrium point $p_c \\in H^ -1 (0)$ and we assume that $p_c$ lies on a strictly convex singular set $S_0 \\subset H^ -1 (0)$. Then we prove that the energy levels $H^ -1 (E)$, with $E>0$ sufficiently small, contain a closed $3$-ball $S_E$ near $S_0$ admitting a system of transverse sections $F_E$, called a $2-3$ foliation. $F_E$ is a singular foliation of $S_E$ and its singular set consists of two periodic orbits $P_{2,E}\\subset \\partial S_E$ and $P_{3,E}\\subset S_E\\setminus \\partial S_E$. The orbit $P_{2,E}$ is hyperbolic inside the energy level $H^ -1 (E)$, lies on the center manifold of the saddle-center $p_c$, has Conley-Zehnder index $2$ and is the asymptotic limit of two rigid planes of $F_E$, which compose the $2$-sphere $S_E$ together with $P_{2,E}$. The orbit $P_{3,E}$ has Conley-Zehnder index $3$ and is the asymptotic limit of a one parameter family of planes of $F_E$ contained in $S_E \\setminus \\partial S_E$. A rigid cylinder connecting the orbits $P_{3,E}$ and $P_{2,E}$ completes the foliation $F_E$. Since $F_E$ is a system of transverse sections, all its regular leaves are transverse to the Hamiltonian flow of $H$. As a consequence of the existence of such foliation in $S_E$, we conclude that the hyperbolic orbit $P_{2,E}$ admits at least one homoclinic orbit contained in $S_E\\setminus \\partial S_E$.