Abstract We present a new variational characterization of breather solutions of any equation of the focusing Gardner hierarchy. This hierarchy is characterized by a nonnegative index n, and 2n + 1 represents the order of the corresponding PDE member. In this paper, we first show the existence of such breathers, and that they are solutions of the (2n+1)th-order Gardner equation. Then we prove a variational universality property, in the sense that all these breather solutions satisfy the same fourth order stationary elliptic ODE, regardless the order of the hierarchy member. This fact also characterizes them as critical points of the same Lyapunov functional, that we also construct here. As by product of our approach, we find breather solutions of the hierarchy of (2n+1)th-order mKdV equations, as well as a respective characterization of them as solutions of a fourth order stationary elliptic ODE. We also extend part of these results to the periodic setting, presenting new breather solutions for the 5th and 7th mKdV members of the hierarchy. Finally, we prove ill-posedness results for the whole Gardner hierarchy, by using appropiately their breather solutions.

Abstract This paper concerns the asymptotic properties of solutions of a class of third-order neutral differential equations with delay. We give sufficient conditions for every solution to be converges to zero, bounded and square integrable. An example is also given to illustrate the results.

Abstract In this paper, we introduce anti-invariant ξ⊥-Riemannian submersions from Hyperbolic β-Kenmotsu Manifolds onto Riemannian manifolds. Necessary and sufficient conditions for a special anti-invariant ξ⊥-Riemannian submersion to be totally geodesic are studied. Moreover, we obtain decomposition theorems for the total manifold of such submersions.

Resumen En este artículo se introducen las submersiones ξ⊥-Riemannianas anti-invariantes desde variedades hiperbólicas β-Kenmotsu sobre variedades Riemannianas. Se estudian condiciones necesarias y suficientes para que ciertas submersiones ξ⊥-Riemannianas anti-invariantes especiales sean totalmente geodésicas. Más aún, se obtienen teoremas de descomposión para la variedad total de dichas submersiones.

Neste trabalho abordamos uma aproximação para o problema de três corpos que interagem mutuamente por atração gravitacional, denominada problema circular restrito de três corpos (PCR3C). Tal aproximação é revisada a partir do formalismo e de conceitos da teoria dos sistemas dinâmicos, enfocando fenômenos de ressonância, representações em espaços de fase, mapas de Poincaré e o Teorema KAM. Discutimos aspectos matemáticos de estabilidade do problema, seus pontos de equilíbrio – pontos Lagrangeanos – e como a condição de integrabilidade se aplica. Resultados de simulações computacionais que desenvolvemos são analisados e algumas aplicações são apresentadas, por exemplo, no que se refere à presença de regiões de comportamento caótico na seção de Poincaré, correlacionadas diretamente a vacâncias em cinturões de asteroides e nos anéis dos planetas gasosos do nosso sistema solar, bem como a possíveis transferências orbitais com menor custo energético.

In this work we present the approximation of the circular restricted three-body problem that interact by gravitation force, the so-called circular restricted three-body problem (CR3BP). Such model is reviewed starting from the formalism and concepts from the theory of dynamic systems, focusing on resonance phenomena, state-space representation, Poincare's maps and the KAM theorem. We have discussed mathematic stability aspects of the problem, its equilibria points – Lagrangean points – and how the integrability condition is applied to do so. Results from computational simulations are analyzed and some are presented, e.g., simulations that explore regions of chaotic behavior on the Poincare's section, directly correlated to the vacancies on asteroids belts and planet's rings from our solar system, as some possible orbit maneuvers that expends less energy.

Abstract In this paper we study topological and analytical conditions on the orbits of a germ of diffeomorphism in the complex plane in order to obtain periodicity. In particular, we give a simple proof of a finiteness criteria for groups of analytic diffeomorphisms, stated in Brochero Martínez 2003. As an application, we derive some consequences about the integrability of complex vector fields in dimension three in a neighborhood of a singular point.

Neste trabalho revisamos alguns dos sistemas não-lineares mais paradigmáticos e desvendamos algumas das suas surpreendentes interligações. Os problemas de interesse, descrito matematicamente pelas equações de sine-Gordon, Toda e KdV, generalizam modelos físicos conhecidos, como o pêndulo simples, o sistema massa-mola e as ondas lineares em água, respectivamente. Depois de discutirmos as peculiaridades decorrentes da presença de não-linearidades nos modelos, esclarecemos como os sistemas apresentados são relacionados uns aos outros, indicando a existência de uma família de equações que compartilham propriedades como a integrabilidade. Nós mostramos como a equação de KdV pode ser convenientemente discretizada a fim de preservar tais características importantes. Além de apresentarmos as ligações estreitas que esta equação tem com a cadeia de Toda e com a equação sine-Gordon, nós também investigamos outros procedimentos capazes de gerar um sistema integrável discreto a partir do modelo KdV com a discretização de Hirota.

In this report we review some of the most paradigmatic nonlinear systems and unveil some of their suprising interconnections. The problems of interest, described mathematically by the equations of sine-Gordon, Toda and KdV, generalize well known physical models, the simple pendulum, the mass on a spring and the linear waves, respectively. After discussing the differences arising from the presence of nonlinearities in the models, we clarify how the systems presented are related to each other, indicating existence of a family of equations sharing integrability properties. We show how the KdV equation can be conveniently discretized in order to preserve important properties. Besides presenting the close connections this equation has with respect to the Toda lattice and the sine-Gordon equation, we also investigate other procedures capable of generating a discrete integrable system from the KdV model, as the Hirota discretization.

The solutions of the Laplace-Beltrami equation on a spherical surface are constructed by the method of separation of variables, as the products of the Fourier basis functions of the azimuthal angle and the integer powers of tangent or cotangent functions of half the polar angle. The Legendre operator acting on the latter functions yields zero. The construction of the Green’s function as the solution of the corresponding Poisson-Beltrami equation with a unit point source on the spherical surface is also constructed using the two-dimensional spherical harmonic basis.

Sufficient conditions for the boundedness and square integrability of solutions and their derivatives of certain fourth order nonlin-ear differential equation are given by means of the Lyapunov’s second method. Our results obtained in this work, generalize existing results on fourth order nonlinear differential equations in the literature. For illustration, an example is also given.

The object of the present paper is to study submanifolds of (k,μ)-contact manifolds. We find the necessary and sufficient conditions for a submanifolds of (k,μ)-contact manifolds to be invariant and anti-invariant. Also, we examine the integrability of the distributions involved in the definition of CR-submanifolds of (k,μ)-contact manifolds.

El objeto del presente artículo es estudiar subvariedades de variedades (k,μ)-contacto. Encontramos las condiciones necesarias y suficientes para que subvariedades de variedades (k,μ)-contacto sean invariantes y anti-invariantes. También examinamos la integrabilidad de las distribuciones involucradas en la definición de subvariedades CR de variedades (k,μ)-contacto.

Sistemas mecânicos sujeitos a vínculos dependentes de velocidades ocorrem com frequência. O teorema de Frobenius permite decidir de forma inequívoca se vínculos que dependem linearmente das velocidades são ou não integráveis. Mostramos como se formula a condição de integrabilidade de Frobenius por meio da teoria das formas diferenciais exteriores e ilustramos sua aplicação a problemas interessantes de mecânica clássica. Por fim, discutimos a importância prática de identificar a natureza dos vínculos no que concerne à integrabilidade.

Mechanical systems subjected to velocity-dependent constraints occur frequently. The Frobenius theorem allows us to unequivocally decide whether constraints that depend linearly on the velocities are integrable or not. We show how to formulate the Frobenius integrability condition by means of the theory of exterior differential forms, and illustrate its application to interesting problems in classical mechanics. Finally, we discuss the practical importance of identifying the nature of the constraints as concerns integrability.

La evaluación de cambios en el bienestar, desde un enfoque microeconómico, se ha concentrado en el cálculo de la variación equivalente y compensatoria a través del enfoque de integrabilidad. Sin embargo, el uso de esta metodología implica especificar de manera precisa la función de demanda de un bien o servicio. Dentro de las áreas sociales, la aproximación de funciones de demanda presenta varios inconvenientes, por lo que se pretende testear la robustez del algoritmo de Vartia y verificar si, en sectores como el de la educación, esta metodología puede constituirse en un mecanismo alternativo para la medición del bienestar cuando existen cambios en los niveles de precio.

The standard approach in order to assess the changes in welfare is the calculation of the equivalent variation or the compensating variation, which relies in what the literature has been called the integrability method. This methodology needs a precise and well defined demand function. In general the use of empirical approximations for this demand function has many drawbacks. In this paper we test the robustness ofVartia algorithm instead of applying the integrability method. We verify that, in sectors like education, this algorithm can be used as an alternative mechanism to assess the changes in the welfare when the level of prices changes.

Sistemas Hamiltonianos quase integráveis são de grande interesse em diversos campos de pesquisa da física e da matemática. Nesses sistemas, o espaço de fase apresenta trajetórias regulares e caóticas. Essas trajetórias dependem, em parte, da amplitude da perturbação que quebra a integrabilidade do sistema. O valor da perturbação crítica responsável por esta transição é um elemento chave no controle do caos. No presente trabalho, exploramos um procedimento para o controle do caos em sistema hamiltoniano quase integrável baseado na alteração de um parâmetro (perturbação). Inicialmente, apresentamos as ferramentas básicas para este estudo: mapa hamiltoniano, linearização do mapa e critério de Chirikov. Posteriormente, investigamos o comportamento de uma interação do tipo onda-partícula frente a perturbação. Por fim, confrontamos os resultados analíticos com uma abordagem numérica (iteração do mapa), mostrando um bom acordo.

Nearly integrable Hamiltonian systems are of great interest in many research fields of physics and mathematics. In these systems, the phase space has regular and chaotic trajectories. These trajectories depend in part on the magnitude of the perturbation that breaks the integrability of the system. The value of the critical perturbation responsible for this transition is a key element in the control of chaos. In this paper, we explore a procedure for the control of chaos in nearly integrable Hamiltonian system based on a parameter change (perturbation). Initially, we present the basic tools for this study: Hamiltonian map, linearization map and Chirikov criterion. Subsequently, we investigated the behavior of a wave-particle interaction type front perturbation. Finally, we confront the analytical approach with a numerical (iteration of the map) results, showing a good agreement.

Se utiliza un enfoque algebraico basado en la descomposión de grupos para mostrar la integrabilidad de un sistema de infinitas ecuaciones de Lax con doble corchete.

A group factorization approach is used to show the integrability of a system of infinite equations of Lax type with double bracket.

As redes de Petri (RPs) constituem-se em uma ferramenta matemática e gráfica de uso geral. Sugere-se, neste trabalho, o uso de RPs como uma ferramenta de modelagem voltada para o ensino de noções matemáticas. A modelagem apoia-se no fato de as RPs terem como característica a representação de sistemas que utilizam forte base matemática. Como estudo de caso, apresenta-se uma modelagem para a organização do conteúdo de ensino sobre integrabilidade de funções reais baseada na evolução do conceito de integral. Para isto, considera-se uma função limitada, definida em um intervalo fechado e limitado da reta, que é analisada relativamente a condições que garantam ou excluam a sua integrabilidade por Cauchy, por Riemann, por Lebesgue ou por nenhum destes. A ideia de se modelar este conceito por RPs originou-se no contexto da sala de aula, por ocasião de seu ensino a alunos de primeiro ano de engenharia.

Petri Nets (PNs) are a mathematical and graphical tool for general use. In this paper, the use of PNs is promoted as a modeling tool to organize the teaching of mathematical notions. The modeling presented in this work relies on the strong mathematical basis of PNs. As a case study, a model is constructed for teaching integrability of real functions, based on the evolution of the integral concept. With this aim, it is considered a real bounded function, defined on a closed and bounded interval, that is analyzed according to conditions which ensure or exclude its integration by Cauchy, by Riemann, by Lebesgue or by none of these integrals. The idea of modeling integrals using PN was originated in the context of mathematical classes, as the integral was being taught to first year engineering students.

The present paper is devoted to obtaining some Ostrowski type inequalities for interval-valued functions. In this context we use the generalized Hukuhara derivative for interval-valued functions. Also some examples and consequences are presented. Mathematical subject classification: Primary: 26E25; Secondary: 35A23.