Fractais e sistemas dinâmicos
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Data
2022-09-02
Autores
Silva, Thiago Martins
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Editor
Universidade Estadual Paulista (Unesp)
Resumo
Fractais são estruturas matemáticas que surgiram para definir formas existentes na realidade e que a geometria clássica não é capaz de descrever, como nuvens e árvores, por exemplo. Estes, não são polígonos, objetos circulares ou nenhuma outra forma que se é conhecida, eles são algo a mais, um fractal, como denominou Mandelbrot. Porém, a teoria sobre essas formas diferentes não se resume a isso, há uma fundamentação matemática bem formalizada em Sistemas Dinâmicos que vai além da Geometria Fractal, suas aplicações são inúmeras e em diversas áreas. Assim, esta dissertação estuda a parte teórica de fractais a partir de Sistemas de Funções Iteradas para, no final, estudar algumas aplicações dessa teoria.
Fractals are Mathematical Structures which arise to define forms that exist in reality and classical geometry was not capable of describing it, like clouds and trees, as an example. These are not polygons, circular objects, nor any other forms that are known, they are something else, a Fractal, as Mandelbrot denominated it. However, the theory about these forms is not restricted only to that, there is a Mathematical fundamentation well formalized in Dynamical Systems going beyond Fractal Geometry, their applications are countless and in many areas. Therefor, this dissertation study the theoretical part of fractals made by Iterated Function System to, in the final, study some applications of the theory.
Fractals are Mathematical Structures which arise to define forms that exist in reality and classical geometry was not capable of describing it, like clouds and trees, as an example. These are not polygons, circular objects, nor any other forms that are known, they are something else, a Fractal, as Mandelbrot denominated it. However, the theory about these forms is not restricted only to that, there is a Mathematical fundamentation well formalized in Dynamical Systems going beyond Fractal Geometry, their applications are countless and in many areas. Therefor, this dissertation study the theoretical part of fractals made by Iterated Function System to, in the final, study some applications of the theory.
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Palavras-chave
Fractal, Sistema dinâmico, IFS, Dimensão fractal, Dynamical system, Fractal dimension