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Master's Dissertation
DOI
10.11606/D.55.2015.tde-07082015-113917
Document
Author
Full name
Alex Pereira da Silva
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2015
Supervisor
Committee
Carvalho, Alexandre Nolasco de (President)
Tahzibi, Ali
Weber, Joachim
Title in Portuguese
Um estudo da teoria das dimensões aplicado a sistemas dinâmicos
Keywords in Portuguese
Dimensão de Hausdorff
Dimensão fractal
Projeção de atrator global
Semigrupo
Abstract in Portuguese
Este trabalho se propõe a estudar o comportamento assintótico dos sistemas dinâmicos autônomos respaldado na Teoria das Dimensões. Mais precisamente, vamos compreender de que maneira nos é útil limitar a dimensão fractal do atrator global de um semigrupo a fim de estudar a dinâmica em dimensão finita, sem que se perca informações sobre a dinâmica ao fazê-lo. Para tanto, o Teorema de Mañé tem um papel decisivo junto às propriedades da dimensão de Hausdorff e a da dimensão fractal; nos permitindo encontrar uma projeção cuja restrição ao atrator é injetora sobre um espaço de dimensão finita. Constatamos ainda que esta abordagem por projeções se aplica largamente a semigrupos originados de equações diferenciais em espaços de Banach de dimensão infinita.
Title in English
A study of dimension theory applied to dynamical system
Keywords in English
Fractal dimension
Hausdorff dimension
Projection of global attractor
Semigroup
Abstract in English
In this work, we study the asymptotic behavior of autonomous dynamical systems supported on the Dimension Theory. More precisely, we understand how fractal dimension finiteness of the global attractor of a semigroup can be used to study the dynamics in finite dimension, without losing information on the dynamics in doing so. For this purpose, the Mañés Theorem plays a decisive role considering the Hausdorff dimension properties and the fractal dimension; thanks to which we managed to find a projection whose restriction to the attractor is an injective application over a finite dimensional space. Besides, we also acknowledge that this projections approach is largely applied to semigroups arrising from differential equations in infinite dimensional Banach spaces.
 
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Publishing Date
2015-08-07
 
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