Abstract Poincaré in 1891 asked about the necessary and sufficient conditions in order to characterize when a polynomial differential system in the plane has a rational first integral. Here we solve this question for the class of Liénard differential equations x ¨ + f ( x ) x ˙ + x = 0, being f ( x ) a polynomial of arbitrary degree. As far as we know it is the first time that all rational first integrals of a relevant class of polynomial differential equations of arbitrary degree has been classified.

Abstract: We present two new classes of polynomial maps satisfying the real Jacobian conjecture in ℝ 2. The first class is formed by the polynomials maps of the form (q(x)–p(y), q(y)+p(x)) : R 2 ⟶ R 2 such that p and q are real polynomials satisfying p'(x)q'(x) ≠ 0. The second class is formed by polynomials maps (f, g): R 2 ⟶ R 2 where f and g are real homogeneous polynomials of the same arbitrary degree satisfying some conditions.

Abstract: This work is based on the comparison of three techniques for analyzing the qualitative behaviour of nonlinear dynamic systems, including the study of their finite and infinite equilibrium points. The qualitative techniques used are: the direct method of Lyapunov, the theorems of Dickson and Perko for second order quadratic differential systems and the linearization around finite equilibrium points. These techniques provide information about the local or global stability of nonlinear systems. The state feedback controlled Buck-Boost power converter will be used as a case of study.

Resumen: Este trabajo se basa en la comparación de tres técnicas de análisis del comportamiento cualitativo de sistemas dinámicos no lineales, incluyendo el estudio de los puntos de equilibrio finitos e infinitos. Las técnicas cualitativas utilizadas son: el Método Directo de Lyapunov, los Teoremas de Dickson y Perko para sistemas cuadráticos de segundo orden y la Linealización alrededor de los puntos de equilibrio finitos. Estas técnicas aportan información respecto a la estabilidad global o local del sistema no lineal. Como sistema dinámico no lineal se utilizará el convertidor de potencia Buck-Boost realimentado por medio del vector de estados.

Abstract In this paper we describe the global phase portrait of the Hamiltonian system associated to a Pinchuk map in the Poincaré disc. In particular, we prove that this phase portrait has 15 separatrices, five of them singular points, and 7 canonical regions, six of them of type strip and one annular.

Resumo Para ε ≠ 0suficientemente pequeno encontramos condições suficientes que garantem a existência de soluções periódicas para sistemas diferenciais de Lienard da forma x ′′ + f ( x ) x ′ + n 2 x + g ( x ) = ε 2 p 1 ( t ) + ε 3 p 2 ( t ) , onde n é um inteiro positivo, f : ℝ → ℝé uma função C 3, g : ℝ → ℝé uma função C 4, e p i : ℝ → ℝpara i = 1 , 2são funções contínuas 2 π–periódicas. A principal ferramenta usada neste artigo é a teoria ’’averaging’’ de segunda ordem. Uma aplicação do resultado principal é feita.

Abstract For ε ≠ 0sufficiently small we provide sufficient conditions for the existence of periodic solutions for the Lienard differential equations of the form x ′′ + f ( x ) x ′ + n 2 x + g ( x ) = ε 2 p 1 ( t ) + ε 3 p 2 ( t ) , where n is a positive integer, f : ℝ → ℝis a C 3function, g : ℝ → ℝis a C 4function, and p i : ℝ → ℝfor i = 1 , 2are continuous 2 π–periodic function. The main tool used in this paper is the averaging theory of second order. We also provide one application of the main result obtained.

Seja F = (f, g) : R2 → R2 uma aplicação polinomial tal que det DF(x) é diferente de zero para todos x ∈ R2. Assumimos que os graus de f e g são iguais. Denotamos por e as partes homogêneas de maior grau de f e g, respectivamente. Nesta nota, damos uma demonstração baseada na teoria qualitativa de equações diferenciais do seguinte resultado: Se não têm fatores lineares em comum, então F é injetora.

Let F= (f, g) : R2 → R2be a polynomial map such that det DF(x) is different from zero for all x∈ R2. We assume that the degrees of fand gare equal. We denote by the homogeneous part of higher degree of f and g, respectively. In this note we provide a proof relied on qualitative theory of differential equations of the following result: If do not have real linear factors in common, then F is injective.

We consider one-parameter families of 2-dimensional vector fields Xµ having in a convenient region R a semistable limit cycle of multiplicity 2m when µ = 0, no limit cycles if µ < 0, and two limit cycles one stable and the other unstable if µ > 0. We show, analytically for some particular families and numerically for others, that associated to the semistable limit cycle and for positive integers n sufficiently large there is a power law in the parameter µ of the form µn ≈ Cnα< 0 with C, α ∈ R, such that the orbit of Xµn through a point of p ∈ R reaches the position of the semistable limit cycle of X0 after given n turns. The exponent α of this power law depends only on the multiplicity of the semistable limit cycle, and is independent of the initial point p ∈ R and of the family Xµ. In fact α = -2m/(2m - 1). Moreover the constant C is independent of the initial point p ∈ R, but it depends on the family Xµ and on the multiplicity 2m of the limit cycle Γ.

Neste trabalho estudamos uma classe de campos vetoriais polinomiais com simetria, definidos no R³ e dependendo de um parâmetro real épsilon, que possui um conjunto de retas invariantes paralelas que tendem para dois pontos singulares no infinito, formando ciclos heteroclínicos degenerados. A análise global na vizinhança dos pontos no infinito é desenvolvida utilizando-se a compactificação de Poincaré. Provamos que para todo n <FONT FACE=Symbol>Î</FONT> N existe épsilonn > 0 tal que, para todo 0 < épsilon < épsilonn, o sistema considerado possui pelo menos n órbitas periódicas de grande amplitude, que bifurcam do ciclo heteroclínico formado pelas duas retas invariantes mais próximas do eixo-x, uma contida no semi-espaço y > 0 e a outra contida no semi-espaço y < 0.

In this paper we study a class of symmetric polynomial differential systems in R³, which has a set of parallel invariant straight lines, forming degenerate heteroclinic cycles, which have their two singular endpoints at infinity. The global study near infinity is performed using the Poincaré compactification. We prove that for all n <FONT FACE=Symbol>Î</FONT> N there is epsilonn > 0 such that for 0 < epsilon < epsilonn the system has at least n large amplitude periodic orbits bifurcating from the heteroclinic loop formed by the two invariant straight lines closest to the x-axis, one contained in the half-space y > 0 and the other in y < 0.

Neste trabalho são determinadas as curvaturas principais e as linhas de curvatura principal das superfícies canal que são as envoltórias de famílias de esferas com raios variáveis e centros deslocando-se ao longo de curvas fechadas regulares em R³. A partir de uma conexão com as equações de Riccati mostra-se que estas superfícies têm no máximo duas curvas principais periódicas isoladas. São dados exemplos com duas curvas principais periódicas simples e com uma dupla.

In this paper are determined the principal curvatures and principal curvature lines on canal surfaces which are the envelopes of families of spheres with variable radius and centers moving along a closed regular curve in R³. By means of a connection of the differential equations for these curvature lines and real Riccati equations, it is established that canal surfaces have at most two isolated periodic principal lines. Examples of canal surfaces with two simple and one double periodic principal lines are given.