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Dinâmica De Partículas Inerciais Acopladas a Um Meio Viscoso ANO 2019 No presente trabalho nós abordamos o estudo da dinâmica do Mapa de MAGALHÃESWILLAMES DE FERREIRA Rede Modificado. Este sistema é obtido pelo método bailout embedding aplicado ao Mapa de Rede, que é Hamiltoniano. O método em questão é formulado ao se considerar o movimento de partículas esféricas num meio viscoso, sendo este o responsável pela dissipação do sistema. Matematicamente a característica do método é transformar um sistema UNIVERSIDADE DO ESTADO DE SANTA CATARINA – UDESC conservativo num dissipativo para se perceber a presença de atratores e, CENTRO DE CIÊNCIAS TECNOLÓGICAS – CCT consequentemente, o surgimento de alguma estabilidade no sistema. Para PROGRAMA DE PÓS-GRADUAÇÃO EM FÍSICA – PPGF a caracterização do Mapa de Rede Modificado, utilizamo-nos das seguintes ferramentas da Dinâmica Não Linear e Mecânica Estatística: Espaço de Fases, Diagramas de Lyapunov, Séries Temporais, Espectro de Lyapunov, Coeficiente de Difusão (D) e Expoente de Difusão. Os resultados obtidos foram: a detecção de regularidade, caos, hipercaos e caos transiente. Além disso, nós abordamos acerca do limite conservativo do método bailout VISCOSO A MEIO ACOPLADAS DINÂMICADEPARTÍCULAS INERCIAIS UM embedding, apesar de não caracterizá-lo, sendo que ainda não há uma abordagem sobre este assunto. O que há de inovador neste trabalho é a apresentação de uma nova forma de se estudar a dispersão das trajetórias no espaço de fases de um sistema dissipativo quadrimensional por meio de DISSERTAÇÃO DE MESTRADO D, observando o comportamento das curvas e constatando que tais modificações são devidas ao surgimento de atratores no sistema. DINÂMICA DE PARTÍCULAS INERCIAIS ACOPLADAS A UM MEIO VISCOSO Orientador: César Manchein JOINVILLE, 2019 WILLAMES FERREIRA DE MAGALHÃES JOINVILLE, 2019 WILLAMES FERREIRA DE MAGALHÃES DINÂMICA DE PARTÍCULAS INERCIAIS ACOPLADAS A UM MEIO VISCOSO Dissertação de mestrado apresentada ao Programa de Pós-Graduação em Física do Centro de Ciências Tecnológicas da Universidade do Estado de Santa Catarina, como requisito parcial para obtenção do grau de Mestre em Física. Orientador: Prof. Dr. César Manchein. Joinville-SC 2019 Ficha catalográfica elaborada pelo programa de geração automática da Biblioteca Setorial do CCT/UDESC, com os dados fornecidos pelo(a) autor(a) Magalhães, Willames Ferreira de Dinâmica de partículas inerciais acopladas a um meio viscoso / Willames Ferreira de Magalhães. -- 2019. 96 p. Orientador: César Manchein Dissertação (mestrado) -- Universidade do Estado de Santa Catarina, Centro de Ciências Tecnológicas, Programa de Pós-Graduação em Física, Joinville, 2019. 1. Mapa de Rede. 2. Mapa de Rede Modificado. 3. Bailout Embedding. 4. Diagramas de Lyapunov. 5. Difusão. I. Manchein, César. II. Universidade do Estado de Santa Catarina, Centro de Ciências Tecnológicas, Programa de Pós-Graduação em Física. III. Título. Dedico este trabalho aos meus pais e a memória de meu avô, Leonardo Magalhães. Agradecimentos Agradeço, inicialmente, por tudo e ainda mais à minha mãe, pela pessoa incrível, inspiradora e humana que é. Ao meu pai, pelos conselhos e ensinamentos da vida. Sou grato à minha irmã, pela cumplicidade de sempre e por sua presença constante, mesmo que de longe. Agradeço ao meu cunhado, pela amizade e conversas proveitosas. A gratidão também se estende à minha sobrinha, que nasceu para trazer mais alegria aos membros família. Além disso, agradecerei eternamente aos meus avós paternos (in memoriam) por eu ser um Magalhães. Sou extremamente grato ao meu orientador, Professor César Manchein, um cientista habilidoso e cheio de ideias originais que me permitiu trabalhar ao seu lado, e esta é uma experiência que jamais me esquecerei. Além disso, agradeço por todas as disponibilidades que teve para comigo ao longo destes dois anos. Agradeço a todos os Professores que tive na vida, pois reconheço que foram eles os responsáveis pela minha chegada até aqui. Gostaria de mencionar alguns dos que tiveram uma contribuição significativa na minha formação: Professor Alcindo Teles Galvão, Professor André de Lima Moura, Professor César Manchein, Professor Fabio Marcel Zanetti, Professor Moreno Pereira Bonutti e Professor Paulo Cesar Rech. Gostaria de agradecer ao Professor Marcus Werner Beims, por algumas disponibili- dades que teve para comigo, e disso jamais me esquecerei. Agradeço a todos os que direta ou indiretamente me ajudaram quando precisei, em especial ao Anderson Hoff. Também sou grato ao Professor Edgard Pacheco Moreira Amorim, à técnica Susele Mussoi Rodrigues e ao técnico Sidnei da Costa Otero pela competência profissional e disponibilidade nos processos administrativos. Agradeço a CAPES, pelo financiamento dos meus estudos; a UDESC pelos custeios durante minhas apresentações em congressos e ao CNPq. “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?”. Edward Norton Lorenz Resumo No presente trabalho nós abordamos o estudo da dinâmica do Mapa de Rede Modificado. Este sistema é obtido pelo método bailout embedding aplicado ao Mapa de Rede, que é Hamiltoniano. O método em questão é formulado ao se considerar o movimento de partículas esféricas num meio viscoso, sendo este o responsável pela dissipação do sistema. Matematicamente a característica do método é transformar um sistema conservativo num dissipativo para se perceber a presença de atratores e, consequentemente, o surgimento de alguma estabilidade no sistema. Para a caracterização do Mapa de Rede Modificado, utilizamo-nos das seguintes ferramentas da Dinâmica Não Linear e Mecânica Estatística: Espaço de Fases, Diagramas de Lyapunov, Séries Temporais, Espectro de Lyapunov, Coeficiente de Difusão (D) e Expoente de Difusão. Os resultados obtidos foram: a detecção de regularidade, caos, hipercaos e caos transiente. Além disso, nós abordamos acerca do limite conservativo do método bailout embedding, apesar de não caracterizá-lo, sendo que ainda não há uma abordagem sobre este assunto. O que há de inovador neste trabalho é a apresentação de uma nova forma de se estudar a dispersão das trajetórias no espaço de fases de um sistema dissipativo quadrimensional por meio de D, observando o comportamento das curvas e constatando que tais modificações são devidas ao surgimento de atratores no sistema. Palavras-chave: Mapa de Rede. Mapa de Rede Modificado. Bailout Embedding. Diagra- mas de Lyapunov. Difusão. Abstract In the present work we study the dynamics of the Modified Web Map. This system is obtained by the bailout embedding method applied to the Web Map, which is Hamil- tonian. The method used is formulated by considering the movement of a particle in a viscous medium which is responsible for the dissipation of the system. Mathematically the characteristic of the method is to transform a conservative system into a dissipative one to detect a presence of attractors and, consequently, the emergence of some stability in the system. For the characterization of the Modified Web Map, the following Nonlinear Dynamics and Statistical Mechanics tools are used: Phase Space, Lyapunov Diagrams, Time Series, Lyapunov Spectrum, Diffusion Coefficient (D) and Diffusion Exponent. The results were: the detection of regularity, chaos, hyperchaos and transient chaos. Also, we approached the conservative limit of the bailout embedding method, although it does not characterize it, and there is still no approach on this subject. What is innovative in this work is a presentation of a new way of studying the dispersion of trajectories in the space of a four-dimensional dissipative system by means of D, observing the behavior of the curves and noting that such modifications are due to the appearance of attractors in the system. Keywords: Web Map. Modified Web Map. Bailout Embedding. Lyapunov Diagrams. Diffusion. viii Lista de abreviações e entidades matemáticas EDO: Equação Diferencial Ordinária. EDO’s: Equações Diferenciais Ordinárias. MR: Mapa de Rede (ou Web Map). MRM: Mapa de Rede Modificado. R: Conjunto dos números reais. Z: Conjunto dos números inteiros. Relação Trigonométrica Fundamental: sen 2(x) + cos2(x) = 1. da dt ≡ a˙, sendo a uma variável qualquer. ix Lista de Figuras Figura 1 – Representações esquemáticas e fora de escala (a) do modelo geocêntrico determinado por Ptolomeu, em que a Terra está no centro do Universo e os corpos celestes (Sol, Lua e outros planetas) giram ao seu redor descrevendo órbitas circulares perfeitas, estando fixas as estrelas e (b) a ilustração do deferente (círculo no qual o planeta se move em torno da Terra) e epiciclo (pequeno círculo em torno do deferente). 15 Figura 2 – Representação de um volume de condições iniciais V (t) no espaço de fases limitado por uma superfície S(t). O vetor gˆ é unitário e F~ designa um campo vetorial tangente a um ponto em S(t)............. 22 Figura 3 – Representação de um elemento de volume de condições iniciais no espaço de fases após uma evolução temporal. Na figura dS simboliza um elemento infinitesimal de área, que após passado um tempo infinitesimal dt, adquire um elemento de volume com base dS e altura (ˆg · F~ )dt. 23 Figura 4 – Representação da intersecção da linha de fluxo C que geram os pontos A e B na Seção de Poincaré bidimensional S, dentro de um espaço tridimensional. ............................... 25 Figura 5 – Gráficos associados à dinâmica do Mapa Logístico xn+1 = (1 − xn)rxn, sendo (a) o Diagrama de Bifurcação e (b) o Expoente de Lyapunov. As √ retas em cinza estão situadas em r = 3, r = 1 + 6 e r = 3, 55424: nos pontos que designam bifurcação em (a) e expoentes de Lyapunov nulos em (b), respectivamente........................... 29 Figura 6 – Diagrama de Lyapunov α × γ para o MRM, tal que 0 ≤ α ≤ 10 e 0 ≤ γ ≤ 22, com uma malha de 1000 × 1000 pontos igualmente espaçados. Uma descrição mais detalhada deste diagrama pode ser encontrada na Figura 15........................... 30 Figura 7 – Representação dos impulsos gerados no sistema.............. 35 Figura 8 – O espaço de fases do Mapa de Rede para (a-d) q = 4; (a-b) K = 1, 5; (c) K = 3, 15 e (d) K = 6, 7. Estas figuras foram construídas utilizando 200 condições iniciais de (v0, u0) escolhidas aleatoriamente no intervalo [0, 2π] com um tempo total de 105 iterações...............
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