Sistemas dinâmicos e caos
Autor(a) principal: | |
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Data de Publicação: | 2023 |
Tipo de documento: | Dissertação |
Idioma: | por |
Título da fonte: | Repositório Institucional da UFS |
Texto Completo: | https://ri.ufs.br/jspui/handle/riufs/18062 |
Resumo: | It was through the historical development of astronomy that the Theory of Dynamical Systems, whose field of study is open and recent, emerged. In this work, we will understand this emergence, learning about important concepts concerning the area. We consider dynamical systems defined by successive applications of a function that maps an interval of real numbers to itself and we study the dynamics of some mathematical models. Among which we can highlight simple examples of financial mathematics, which is a branch very close to our reality, and the study of the tent function. We introduce the notion of equivalence between dynamical systems defined by iteration of functions and, through this notion, we get to know the dynamics of new systems. We also study asymptotic stability of a fixed point or periodic point of a dynamical system. We present the topological definition of chaos and discuss some essential features of this important concept. We analyze again the tent function and present, through the binary expansion of real numbers in the interval [0, 1], a proof that the dynamical system defined by this function and, consequently, any other equivalent to it, is chaotic. Finally, we examine the "logistic population model" discussed by May( [6]), highlighting some of its features. |
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Cabral, Bruno da SilvaVeiga, Ana Cristina Salviano2023-08-08T00:24:16Z2023-08-08T00:24:16Z2023-05-26CABRAL, Bruno da Silva. Sistemas dinâmicos e caos. 2023. 67 f. Dissertação (Mestrado Profissional em Matemática) - Universidade Federal de Sergipe, São Cristóvão, 2023.https://ri.ufs.br/jspui/handle/riufs/18062It was through the historical development of astronomy that the Theory of Dynamical Systems, whose field of study is open and recent, emerged. In this work, we will understand this emergence, learning about important concepts concerning the area. We consider dynamical systems defined by successive applications of a function that maps an interval of real numbers to itself and we study the dynamics of some mathematical models. Among which we can highlight simple examples of financial mathematics, which is a branch very close to our reality, and the study of the tent function. We introduce the notion of equivalence between dynamical systems defined by iteration of functions and, through this notion, we get to know the dynamics of new systems. We also study asymptotic stability of a fixed point or periodic point of a dynamical system. We present the topological definition of chaos and discuss some essential features of this important concept. We analyze again the tent function and present, through the binary expansion of real numbers in the interval [0, 1], a proof that the dynamical system defined by this function and, consequently, any other equivalent to it, is chaotic. Finally, we examine the "logistic population model" discussed by May( [6]), highlighting some of its features.Foi por meio do desenvolvimento historico da astronomia que se surgiu a Teoria dos Sistemas Dinamicos, cuja area de estudo é aberta e recente. Nesse trabalho, entenderemos esse surgimento, tomando conhecimento sobre conceitos importantes inerentes a essa área. Consideramos sistemas dinamicos definidos por aplicacoes sucessivas de uma funcao que aplica um intervalo de numeros reais nele mesmo e estudamos a dinamica de alguns modelos matematicos. Dentre os quais podemos destacar exemplos simples de matematica financeira, que é um ramo muito proximo da nossa realidade, e o estudo da funcao tenda. Introduzimos a nocao de equivalencia entre sistemas dinamicos definidos por iteracao de funcoes e, por meio dessa nocao, passamos a conhecer a dinamica de novos sistemas. Estudamos ainda estabilidade assintotica de um ponto fixo e de um ponto peri´odico de um sistema dinamico. Apresentamos a definicao topologica de caos e discutimos algumas caracterısticas essenciais desse importante conceito. Analisamos novamente a funcao tenda e apresentamos, por meio da expansao binaria de numeros reais no intervalo [0, 1], uma prova que o sistema dinamico definido por essa funcao e, consequentemente, qualquer outro equivalente a ele, é caotico. Por fim, examinamos o “modelo de populacao logıstica” discutido por May( [6]), destacando algumas de suas caracterısticas.São CristóvãoporSistemas dinâmicosIteração de funçõesEstabilidadeConjugaçãoFunção tendaCaosDynamical systemsIterating functionsStabilityConjugationTent functionChaosCIENCIAS EXATAS E DA TERRA::MATEMATICASistemas dinâmicos e caosinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesisMestrado Profissional em MatemáticaUniversidade Federal de Sergipe (UFS)reponame:Repositório Institucional da UFSinstname:Universidade Federal de Sergipe (UFS)instacron:UFSinfo:eu-repo/semantics/openAccessLICENSElicense.txtlicense.txttext/plain; charset=utf-81475https://ri.ufs.br/jspui/bitstream/riufs/18062/1/license.txt098cbbf65c2c15e1fb2e49c5d306a44cMD51ORIGINALBRUNO_SILVA_CABRAL.pdfBRUNO_SILVA_CABRAL.pdfapplication/pdf1422239https://ri.ufs.br/jspui/bitstream/riufs/18062/2/BRUNO_SILVA_CABRAL.pdf04d9c0e7b09501478c85d012c464c9aeMD52TEXTBRUNO_SILVA_CABRAL.pdf.txtBRUNO_SILVA_CABRAL.pdf.txtExtracted texttext/plain98509https://ri.ufs.br/jspui/bitstream/riufs/18062/3/BRUNO_SILVA_CABRAL.pdf.txt2986b85549ce24081dde406b6a1477a9MD53THUMBNAILBRUNO_SILVA_CABRAL.pdf.jpgBRUNO_SILVA_CABRAL.pdf.jpgGenerated Thumbnailimage/jpeg1229https://ri.ufs.br/jspui/bitstream/riufs/18062/4/BRUNO_SILVA_CABRAL.pdf.jpgdf60eb49c69409e718575bc350a03b46MD54riufs/180622023-08-07 21:24:21.746oai:ufs.br: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Repositório InstitucionalPUBhttps://ri.ufs.br/oai/requestrepositorio@academico.ufs.bropendoar:2023-08-08T00:24:21Repositório Institucional da UFS - Universidade Federal de Sergipe (UFS)false |
dc.title.pt_BR.fl_str_mv |
Sistemas dinâmicos e caos |
title |
Sistemas dinâmicos e caos |
spellingShingle |
Sistemas dinâmicos e caos Cabral, Bruno da Silva Sistemas dinâmicos Iteração de funções Estabilidade Conjugação Função tenda Caos Dynamical systems Iterating functions Stability Conjugation Tent function Chaos CIENCIAS EXATAS E DA TERRA::MATEMATICA |
title_short |
Sistemas dinâmicos e caos |
title_full |
Sistemas dinâmicos e caos |
title_fullStr |
Sistemas dinâmicos e caos |
title_full_unstemmed |
Sistemas dinâmicos e caos |
title_sort |
Sistemas dinâmicos e caos |
author |
Cabral, Bruno da Silva |
author_facet |
Cabral, Bruno da Silva |
author_role |
author |
dc.contributor.author.fl_str_mv |
Cabral, Bruno da Silva |
dc.contributor.advisor1.fl_str_mv |
Veiga, Ana Cristina Salviano |
contributor_str_mv |
Veiga, Ana Cristina Salviano |
dc.subject.por.fl_str_mv |
Sistemas dinâmicos Iteração de funções Estabilidade Conjugação Função tenda Caos |
topic |
Sistemas dinâmicos Iteração de funções Estabilidade Conjugação Função tenda Caos Dynamical systems Iterating functions Stability Conjugation Tent function Chaos CIENCIAS EXATAS E DA TERRA::MATEMATICA |
dc.subject.eng.fl_str_mv |
Dynamical systems Iterating functions Stability Conjugation Tent function Chaos |
dc.subject.cnpq.fl_str_mv |
CIENCIAS EXATAS E DA TERRA::MATEMATICA |
description |
It was through the historical development of astronomy that the Theory of Dynamical Systems, whose field of study is open and recent, emerged. In this work, we will understand this emergence, learning about important concepts concerning the area. We consider dynamical systems defined by successive applications of a function that maps an interval of real numbers to itself and we study the dynamics of some mathematical models. Among which we can highlight simple examples of financial mathematics, which is a branch very close to our reality, and the study of the tent function. We introduce the notion of equivalence between dynamical systems defined by iteration of functions and, through this notion, we get to know the dynamics of new systems. We also study asymptotic stability of a fixed point or periodic point of a dynamical system. We present the topological definition of chaos and discuss some essential features of this important concept. We analyze again the tent function and present, through the binary expansion of real numbers in the interval [0, 1], a proof that the dynamical system defined by this function and, consequently, any other equivalent to it, is chaotic. Finally, we examine the "logistic population model" discussed by May( [6]), highlighting some of its features. |
publishDate |
2023 |
dc.date.accessioned.fl_str_mv |
2023-08-08T00:24:16Z |
dc.date.available.fl_str_mv |
2023-08-08T00:24:16Z |
dc.date.issued.fl_str_mv |
2023-05-26 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/masterThesis |
format |
masterThesis |
status_str |
publishedVersion |
dc.identifier.citation.fl_str_mv |
CABRAL, Bruno da Silva. Sistemas dinâmicos e caos. 2023. 67 f. Dissertação (Mestrado Profissional em Matemática) - Universidade Federal de Sergipe, São Cristóvão, 2023. |
dc.identifier.uri.fl_str_mv |
https://ri.ufs.br/jspui/handle/riufs/18062 |
identifier_str_mv |
CABRAL, Bruno da Silva. Sistemas dinâmicos e caos. 2023. 67 f. Dissertação (Mestrado Profissional em Matemática) - Universidade Federal de Sergipe, São Cristóvão, 2023. |
url |
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Mestrado Profissional em Matemática |
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Universidade Federal de Sergipe (UFS) |
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