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Item Superfícies de Bianchi com ângulo de Chebyshev constante(Universidade Federal de Goiás, 2010-10-01) BEZERRA, Adriano Cavalcante; CORRO, Armando Mauro Vasquez; http://lattes.cnpq.br/4498595305431615The Bianchi surfaces belong to a class of surfaces with negative Gaussian curvature, discovered by generalization of Backlund transformation for surfaces with constant negative Gaussian curvature [3]. Today these areas are studied from the viewpoint of the theory of integrable systems. In this paper we study Bianchi surfaces parameterized by a Generalized Chebyshev net and show that such a surfaces with Chebyshev constant angle is a piece of a right helicoid, see [1].Item Sobre Regularização e Perturbação Singular(Universidade Federal de Goiás, 2011-02-24) CASTRO, Ubirajara José Gama de; MEDRADO, João Carlos da Rocha; http://lattes.cnpq.br/5021927574622286The main goal of this work is to study the behavior of Discontinuous Vector Fields in a neighborhood of a tipical singularity (tangency) using for this the regularization process developed by Teixeira and Sotomayor [9] and, using also, some technics of the Geometric Singular Perturbation Theory [2].Item Superfícies Completas com Curvatura Gaussiana Constante em H2×R e S2×R(Universidade Federal de Goiás, 2010-03-19) CINTRA, Adriana Araujo; PINA, Romildo da Silva; http://lattes.cnpq.br/2675728978857991In this work we classify the complete surfaces with constant Gaussian curvature into the H2×R and S2×R.We show that exists a unique complete surface, up to isometries, with positive constant Gaussian curvature into the H2×R, and greater than one, into the S2×R and that there is no complete surfaces with constant Gaussian curvature K(I) < −1 into the H2×R and S2×R. We prove that even if −1 ≤ K(I) < 0 there are infinite complete surfaces into the H2 ×R with Gaussian curvature K(I) and with additional assumption we prove there is if −1 ≤ K(I) < 0 and 0 < K(I) < 1 there is no exists complete surfaces into S2×R with Gaussian curvature K(I). These results were obtained by Aledo, Espinar and Gálvez and can be found in [1].Item Sobre uma caracterização de pequena calota esférica(Universidade Federal de Goiás, 2011-02-18) DIAS, Diogo Gonçalves; FERREIRA, Walterson Pereira; http://lattes.cnpq.br/9150818921025647It is known that small spherical caps are the only compact surfaces with constant mean curvature H 6= 0 graphics whose boundary is a round circle. This characterization is a partial answer to one of the conjectures of the spherical cap and its classic demonstration involves the Maximum Principle for surfaces with constant mean curvature. What we re doing this work is to give a new proof for this characterization of small spherical cap without the use of the Principle of Maximum. In addition, we make statements alternatives other results related to Conjecture, whenever possible.Item Inflexões de Linhas Assintóticas e de Linhas de Curvatura em Superfícies(Universidade Federal de Goiás, 2010-10-19) FREITAS, Bruno Rodrigues de; GARCIA, Ronaldo Alves; http://lattes.cnpq.br/5680428710939826Quadratic points (or special hyperbolic points) are points where a surface can be approximated by a quadric to the terms of order three. We will deal with a conjecture that asserts that every closed hyperbolic surface in RP3 has not less than eight distinct quadratic points. We prove a result which states that; if a generic surface in RP3 contains a hyperbolic disk bounded by a Jordan parabolic curve, then there is an odd number of quadratic points inside this disc. We study curves formed by the inflection points of asymptotic foliations and principals in the hyperbolic domain.We studied the behavior of the inflection curve of the asymptotically foliation near a special parabolic point (the point where the asymptotic direction is tangent to the parabolic curve), and the behavior of the inflection curve of the principal foliation near a umbilic point.Item Superfícies de Weingarten Lineares Hiperbólicas em R3(Universidade Federal de Goiás, 2009-08-25) GUEDES, Luciene Viana; FERREIRA, Walterson Pereira; http://lattes.cnpq.br/9150818921025647The present work has been based by the [1] from Juan A. Aledo S´anches and Jos´e M. Espinar and [2] from Rafael L´opez articles. In those articles they studied hiperbolic linear Weingarten surfaces in R3 space, this is, surface whose mean curvature H and Gaussian curvature K satisfy a relation of the form aH+bK =c, where a, b, c 2 R. A such surface is said to be hiperbolic when the discriminant D := a2+4bc < 0.We obtain a representation for rotational hyperbolic linear Weingarten surfaces in terms of its Gauss map and we also present, in the case a 6= 0, a classification of linearWeingarten surfaces of hyperbolic rotation. As a consequence we obtain, in the case a 6=0, a family of complete hyperbolic linear Weingarten surfaces in R3. This contrasts with Hilbert s theorem that there do not exist complete surfaces with constant negative Gaussian curvature immersed in R3.Item Par de Curvas no Plano: Geometria da Bicicleta(Universidade Federal de Goiás, 2011-03-25) LEE, Fang Chou; GARCIA, Ronaldo Alves; http://lattes.cnpq.br/5680428710939826The main objective is to study the curves generated by the front and rear wheels of a bicycle from the standpoint of differential geometry.Item Superfícies Regradas de Bonnet(Universidade Federal de Goiás, 2011-03-31) LEITE, Elaine Altino Freire; CORRO, Armando Mauro Vasquez; http://lattes.cnpq.br/4498595305431615In this work we show that a Surface is a Bonnet Surface if, and only if A-net, presenting in Soyuçok s work [6]. Using this result we study the Bonnet Ruled Surfaces, based in Kanbay s work [1].Item O Método do Averaging via Grau de Brouwer para determinar o número de ciclos limites de um centro 4-dimensional em sistemas de controle.(Universidade Federal de Goiás, 2010-03-30) MALAQUIAS, Arianny Grasielly Baiao; MIZUKOSHI, Marina Tuyako; http://lattes.cnpq.br/0850581921280261; MEDRADO, João Carlos da Rocha; http://lattes.cnpq.br/5021927574622286In this work, we studying the Averaging Method via Brouwer Degree for upper bound the number of limit cycle that can bifurcate from a center type singularity of a diferential equation system. After that, we give concrete examples this upper bound can be realized.Item Centros Persistentes(Universidade Federal de Goiás, 2010-03-05) ROCHA, Valdomiro; MEDRADO, João Carlos da Rocha; http://lattes.cnpq.br/5021927574622286The problem of destingnishing whether a monodromic critical point with imaginary eigenvalues of a family of a planar analitical vector field is a center or a focus was already solved by Lyapunov. This is the famous center-focus problem which was solved by calculating the so-called Lyapunov constants and see whether or not they are zero. We present a few ways to calculate them acording the approaches that they use: camputation of a Lyapunov function; use of normal forms; computation of the power of expansion of a solution in polar coordinates; use of the algebraic structure of Lyapunov constants; method of Lyapunov-Schmit and Melnikov functions. Despite all of the above the centerfocus problem for a simple family as the cube is resisting all attempts at solution. For this reason the centers, we propose to grade the in three levels in order to make the problem more feasible.Item Equações Diferenciais por partes:ciclos limite e cones invaiantes(Universidade Federal de Goiás, 2011-03-25) SILVA, Thársis Souza; GARCIA, Ronaldo Alves; http://lattes.cnpq.br/5680428710939826In this work, we consider classes of discontinuous piecewise linear systems in the plane and continuous in the space. In the plane, we analyze systems of focus-focus (FF), focusparabolic (FP) and parabolic-parabolic (PP) type, separated by the straight line x = 0, and we prove that can appear until two limit cycles depending of parameters variations. Also we study a specific system, piecewise, with two saddles (one fixed in the origin and the other in the neighborhood of point (1;1)) separated by the straight line y= -x+1, and we show that can appear until two limit cycles depending of parameters variations. Finally, we examine a continuous piecewise linear system in R³ and we prove the existence of invariant cones and, through this structures, we determine some stable and unstable behavior.Item Transformações de Ribaucour para hipersuperfícies em formas espaciais(Universidade Federal de Goiás, 2008-02-29) SOUTO, Leonardo Antônio; LEMES, Max Valério; http://lattes.cnpq.br/6512955169729670The theory of Ribaucour transformations for hypersurfaces in space forms is presented. A method to obtaining linear Weingarten surfaces in a three-dimensional space form is showed. By applying the theory to the cylinder, we obtain a two-parameter family of linear Weingarten surfaces. A new one-parameter family of complete constant mean curvature surfaces in the unit sphere, locally associated to the flat torus, is obtained. We construct new families of constant mean curvature 1 (cmc-1) surfaces which are locally associated to Enneper cousin.Item Superfícies Helicoidais no espaço Euclidiano e de Minkowski(Universidade Federal de Goiás, 2012-05-31) SOUZA, Danillo Flugge de; PINA, Romildo da Silva; http://lattes.cnpq.br/2675728978857991In this work, based in [2] and [6] we studies helicoidal surfaces of the Euclidean space and Minkowski space R31 with prescribed Gaussian or mean curvature given by smooth functions. In the Minkowski space we consider three especial kinds of helicoidal surfaces, corresponding to the space-like, time-like or light-like axes of revolution and show some geometric meanings of the helicoidal surfaces of the space-like type. We also define certain solinoid (tubular) surfaces around a hyperbolic helix in R31and we study some of their geometric properties.Item Uma condição de injetividade e a estabilidade assintótica global no plano(Universidade Federal de Goiás, 2010-03-29) SOUZA, Wender José de; GARCIA, Ronaldo Alves; http://lattes.cnpq.br/5680428710939826In this work we are interested in the solution of the following problem: Let Y = ( f ,g) be a vector field of class C1 in R2. Suppose that (x, y) = (0,0) is a singular point of Y and assume that for any q ∈ R2, the eigenvalues of DY have negative real part, this is, det(DY) > 0 and tr(DY) < 0. Then, the solution (x, y) = (0,0) of Y is globally asymptotically stable. To this end, we show that this problema is equivalent to the following: Let Y : R2 →R2 be a C1 vector field. If det(DY) > 0 and tr(DY) < 0, then Y is globally injective. This equivalence was proved by C. Olech [1]. So we show the injectivity of the vector field Y under the conditions det(DY) > 0 and tr(DY)<0. In fact, we present a more stronger result, which was obtained by C. Gutierrez and can be found in [4]. This result is given by: Any planar vector field X of class C2 satisfying the r-eigenvalue condition for some r ∈ [0,¥) is injective.Item Superfícies de Weingarten Generalizadas do Tipo Rotacional no 3-Espaço Euclidiano(Universidade Federal de Goiás, 2011-03-01) VELASCO, Lívio José; CORRO, Armando Mauro Vasquez; http://lattes.cnpq.br/4498595305431615In this work, we study the surfaces of rotation S which are Weingarten general, in which the Gaussian curvature K and mean curvature H of this surface satisfies the following relationship (w2 r2)K +2wH +1 = 0, where w and r are harmonic functions with respect to the quadratic form s = II +wIII and II, III are the surface s second and third quadratic form. Inspired by the work of Schief [15], we obtain a characterization of these surfaces determined by functions satisfying a system of ordinary differential equations, as application we prove that with an additional condition these surfaces are spheres.Item Curvatura de Lie das hipersuperfícies de Dupin(Universidade Federal de Goiás, 2008-03-28) VITOR, Erivelton Paulo; RODRIGUES, Luciana Maria Dias de ávila; http://lattes.cnpq.br/6564647402919278In this work we study some results from the article of Tomas E. Cecil, On the Lie curvature of Dupin hypersurfaces [4]. We study the basic concepts of Lie sphere geometry, which given the framework for the study of Dupin hypersurfaces in the Lie sphere geometry. We construct example of a non-compact proper Dupin hypersurface immersed in Sn on which the Lie curvature Ψ = 1/2 which is not Lie equivalent to an open subset of an isoparametric hypersurface in Sn. We also produce example on which Lie curvature Ψ has a constant value c, 0 < c < 1.