Instability of equilibrium points of some Lagrangian systems

In this work we show that, if L is a natural Lagrangian system such that the k-jet of the potential energy ensures it does not have a minimum at the equilibrium and such that its Hessian has rank at least n - 2, then there is an asymptotic trajectory to the associated equilibrium point and so the equilibrium is unstable. This applies, in particular, to analytic potentials with a saddle point and a Hessian with at most 2 null eigenvalues. The result is proven for Lagrangians in a specific form, and we show that the class of Lagrangians we are interested can be taken into this specific form by a subtle change of spatial coordinates. We also consider the extension of this results to systems subjected to gyroscopic forces. (C) 2008 Elsevier Inc. All rights reserved.

On the Hamilton-Jacobi equation in the framework of generalized functions

In this work we study, in the framework of Colombeau`s generalized functions, the Hamilton-Jacobi equation with a given initial condition. We have obtained theorems on existence of solutions and in some cases uniqueness. Our technique is adapted from the classical method of characteristics with a wide use of generalized functions. We were led also to obtain some general results on invertibility and also on ordinary differential equations of such generalized functions. (C) 2011 Elsevier Inc. All rights reserved.

Reaction-diffusion systems coupled at the boundary and the Morse-Smale property

We study an one-dimensional nonlinear reaction-diffusion system coupled on the boundary. Such system comes from modeling problems of temperature distribution on two bars of same length, jointed together, with different diffusion coefficients. We prove the transversality property of unstable and stable manifolds assuming all equilibrium points are hyperbolic. To this end, we write the system as an equation with noncontinuous diffusion coefficient. We then study the nonincreasing property of the number of zeros of a linearized nonautonomous equation as well as the Sturm-Liouville properties of the solutions of a linear elliptic problem. (C) 2008 Elsevier Inc. All rights reserved.

Exploratory semiparametric analysis of two-dimensional diffusions in finance

We examine bivariate extensions of Aït-Sahalia’s approach to the estimation of univariate diffusions. Our message is that extending his idea to a bivariate setting is not straightforward. In higher dimensions, as opposed to the univariate case, the elements of the Itô and Fokker-Planck representations do not coincide; and, even imposing sensible assumptions on the marginal drifts and volatilities is not sufficient to obtain direct generalisations. We develop exploratory estimation and testing procedures, by parametrizing the drifts of both component processes and setting restrictions on the terms of either the Itô or the Fokker-Planck covariance matrices. This may lead to highly nonlinear ordinary differential equations, where the definition of boundary conditions is crucial. For the methods developed, the Fokker-Planck representation seems more tractable than the Itô’s. Questions for further research include the design of regularity conditions on the time series dependence in the data, the kernels actually used and the bandwidths, to obtain asymptotic properties for the estimators proposed. A particular case seems promising: “causal bivariate models” in which only one of the diffusions contributes to the volatility of the other. Hedging strategies which estimate separately the univariate diffusions at stake may thus be improved.

A importância dos mecanismos comportamentais de resistência para a dinâmica populacional de abelhas Apis mellifera e o parasita Varroa destructor

Os ácaros ectoparasitas Varroa destructor, que parasitam as abelhas tornaram-se um problema global. Embora seja pouco provável que estes ácaros, por si só, provoquem a mortalidade das colmeias, eles desempenham um importante papel como vetor de muitas doenças virais. E estas doenças são identificados como algumas das mais importantes razões para a Desordem do Colapso das Colônias. Os efeitos da infestação do V.destructor são distintas em diferentes partes do mundo. Maiores mortalidades de colônias têm sido relatadas em colônias de abelhas européias (AE) em países da Europa, Ásia e América do Norte. No entanto, este ácaro está presente no Brasil já por muitos anos e não existem relatos de perdas em colônias das abelhas africanizadas (AA). Estudos realizados no México mostraram que alguns comportamentos de resistência ao ácaro Varroa - especialmente o grooming e o comportamento higiênico - são diferentes em cada uma das subespécie. Poderiam então esses mecanismos explicar por que as abelhas africanizadas são menos suscetíveis à Desordem do Colapso das Colônias? A fim de responder a esta pergunta, propomos um modelo matemático baseado em equações diferenciais, com o objetivo de analisar o papel desses mecanismos de resistência na saúde geral da colônia e na capacidade da colônia para enfrentar desafios ambientais.

Um método de linearização local com passo adaptativo para solução numérica de equações diferenciais estocásticas com ruído aditivo

Neste trabalho apresentamos um novo método numérico com passo adaptativo baseado na abordagem de linearização local, para a integração de equações diferenciais estocásticas com ruído aditivo. Propomos, também, um esquema computacional que permite a implementação eficiente deste método, adaptando adequadamente o algorítimo de Padé com a estratégia “scaling-squaring” para o cálculo das exponenciais de matrizes envolvidas. Antes de introduzirmos a construção deste método, apresentaremos de forma breve o que são equações diferenciais estocásticas, a matemática que as fundamenta, a sua relevância para a modelagem dos mais diversos fenômenos, e a importância da utilização de métodos numéricos para avaliar tais equações. Também é feito um breve estudo sobre estabilidade numérica. Com isto, pretendemos introduzir as bases necessárias para a construção do novo método/esquema. Ao final, vários experimentos numéricos são realizados para mostrar, de forma prática, a eficácia do método proposto, e compará-lo com outros métodos usualmente utilizados.

Utilização de mercados artificiais com formadores de mercado para análise de estratégias

Na modelagem de sistemas complexos, abordagens analíticas tradicionais com equações diferenciais muitas vezes resultam em soluções intratáveis. Para contornar este problema, Modelos Baseados em Agentes surgem como uma ferramenta complementar, onde o sistema é modelado a partir de suas entidades constituintes e interações. Mercados Financeiros são exemplos de sistemas complexos, e como tais, o uso de modelos baseados em agentes é aplicável. Este trabalho implementa um Mercado Financeiro Artificial composto por formadores de mercado, difusores de informações e um conjunto de agentes heterogêneos que negociam um ativo através de um mecanismo de Leilão Duplo Contínuo. Diversos aspectos da simulação são investigados para consolidar sua compreensão e assim contribuir com a concepção de modelos, onde podemos destacar entre outros: Diferenças do Leilão Duplo Contínuo contra o Discreto; Implicações da variação do spread praticado pelo Formador de Mercado; Efeito de Restrições Orçamentárias sobre os agentes e Análise da formação de preços na emissão de ofertas. Pensando na aderência do modelo com a realidade do mercado brasileiro, uma técnica auxiliar chamada Simulação Inversa, é utilizada para calibrar os parâmetros de entrada, de forma que trajetórias de preços simulados resultantes sejam próximas à séries de preços históricos observadas no mercado.

A Numerical method for the semilinear stochastic transport equation

Trabalho apresentado no XXXV CNMAC, Natal-RN, 2014.

Numerical integration of coupled stochastic oscillators driven by Random Forces

Trabalho apresentado no International Conference on Scientific Computation And Differential Equations 2015

Modelagem e simulação de um sistema de bombeio mecânico em poços direcionais utilizando parâmetros concentrados

This work aims presenting the development of a model and computer simulation of a sucker rod pumping system. This system take into account the well geometry, the flow through the tubing, the dynamic behavior of the rod string and the use of a induction motor model. The rod string were modeled using concentrated parameters, allowing the use of ordinary differential equations systems to simulate it s behavior

Uma reavaliação do pensamento lógico de George Boole à luz da história da matemática

The aim of the present study is to reevaluate the logical thought of the English mathematician George Boole (1815 - 1864). Thus, our research centers on the mathematical analysis of logic in the context of the history of mathematics. In order to do so, we present various biographical considerations about Boole in the light of events that happened in the 19th century and their consequences for mathematical production. We briefly describe Boole's innovations in the areas of differential equations and invariant theory and undertake an analysis of Boole's logic, especially as formulated in the book The Mathematical Analysis of Logic, comparing it not only with the traditional Aristotelian logic, but also with modern symbolic logic. We conclude that Boole, as he intended, expanded logic both in terms of its content and also in terms of its methods and formal elaboration. We further conclude that his purpose was the mathematical modeling of deductive reasoning, which led him to present an innovative formalism for logic and, because the different ways it can be interpreted, a new conception of mathematics

Aplicação de sistemas multi-classificadores no diagnóstico de falhas em motores de indução trifásicos

Equipment maintenance is the major cost factor in industrial plants, it is very important the development of fault predict techniques. Three-phase induction motors are key electrical equipments used in industrial applications mainly because presents low cost and large robustness, however, it isn t protected from other fault types such as shorted winding and broken bars. Several acquisition ways, processing and signal analysis are applied to improve its diagnosis. More efficient techniques use current sensors and its signature analysis. In this dissertation, starting of these sensors, it is to make signal analysis through Park s vector that provides a good visualization capability. Faults data acquisition is an arduous task; in this way, it is developed a methodology for data base construction. Park s transformer is applied into stationary reference for machine modeling of the machine s differential equations solution. Faults detection needs a detailed analysis of variables and its influences that becomes the diagnosis more complex. The tasks of pattern recognition allow that systems are automatically generated, based in patterns and data concepts, in the majority cases undetectable for specialists, helping decision tasks. Classifiers algorithms with diverse learning paradigms: k-Neighborhood, Neural Networks, Decision Trees and Naïves Bayes are used to patterns recognition of machines faults. Multi-classifier systems are used to improve classification errors. It inspected the algorithms homogeneous: Bagging and Boosting and heterogeneous: Vote, Stacking and Stacking C. Results present the effectiveness of constructed model to faults modeling, such as the possibility of using multi-classifiers algorithm on faults classification

Avaliacao de dados de secagem de suspensoes de polpas de frutas em leito de jorro com alimentacao intermitente

The objective of this work was the development and improvement of the mathematical models based on mass and heat balances, representing the drying transient process fruit pulp in spouted bed dryer with intermittent feeding. Mass and energy balance for drying, represented by a system of differential equations, were developed in Fortran language and adapted to the condition of intermittent feeding and mass accumulation. Were used the DASSL routine (Differential Algebraic System Solver) for solving the differential equation system and used a heuristic optimization algorithm in parameter estimation, the Particle Swarm algorithm. From the experimental data food drying, the differential models were used to determine the quantity of water and the drying air temperature at the exit of a spouted bed and accumulated mass of powder in the dryer. The models were validated using the experimental data of drying whose operating conditions, air temperature, flow rate and time intermittency, varied within the limits studied. In reviewing the results predicted, it was found that these models represent the experimental data of the kinetics of production and accumulation of powder and humidity and air temperature at the outlet of the dryer

Resonances and subharmonic bifurcations of large amplitude periodic orbits of planar polynomial vector fields

In this work are studied periodic perturbations, depending on two parameters, of planar polynomial vector fields having an annulus of large amplitude periodic orbits, which accumulate on a symmetric infinite heteroclinic cycle. Such periodic orbits and the heteroclinic trajectory can be seen only by the global consideration of the polynomial vector fields on the whole plane, and not by their restriction to any compact set. The global study involving infinity is performed via the Poincare Compactification. It is shown that, for certain types of periodic perturbations, one can seek, in a neighborhood of the origin in the parameter plane, curves C-(m) of subharmonic bifurcations, for which the periodically perturbed system has subharmonics of order m, for any integer m.

DEGENERATE HOPF BIFURCATIONS IN CHUA'S SYSTEM

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)