Takensov teorem i primjene

master's thesis

• Author: Karla Živković
• Mentor: Siniša Slijepčević (mentor)
• Committee member: Siniša Slijepčević (predsjednik povjerenstva)
• Committee member: Mladen Jurak (član povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Committee member: Bojan Basrak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-09-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Na samom početku rada, u poglavlju Uvod u dinamičke sustave, obradili smo osnovne pojmove, te naveli neke primjere dinamičkih sustava. Objasnili smo kakvi sustavi mogu biti, definirali orbite, vrste točaka sustava i definirali za dinamičke sustave vrlo važan pojam atraktora i neka svojstva bitna za njegovo razumijevanje. U drugom poglavlju proučavali smo topološke dinamičke sustave i naveli neka njihova svojstva. Objasnili smo sto su to granični i minimalni skupovi, termin povratnosti točaka, te kakvi sustavi moraju biti da bi bili topološki tranzitivni, ekspanzivni ili topološko miješanje. Nadalje, u poglavlju Dimenzije, proučavali smo tri različite definicije dimenzije; Lebesgueovu pokrivajuću dimenziju, Hausdorffovu te box dimenziju. Također proučavali smo pojam konačnodimenzionalnosti u terminima teorije ulaganja u Euklidske prostore za svaku od navedenih dimenzija. U četvrtom poglavlju bavili smo se glavnom temom ovoga rada, Takensovim teoremom i njegovim dokazom, te nekim tehničkim rezultatima potrebnima za dokazivanje istoga. U posljednjem, petom poglavlju, Primjena, obradili smo jedan od najvažnijih primjera teorije dinamičkih sustava - Lorenzove jednadžbe i njima pripadajući Lorenzov atraktor.

Abstract (english)

At the very beginning of this paper we are introduced to some basic examples and concepts of dynamic systems. We explained different types of dynamic systems, defined orbits, types of points in the dynamic systems and introduced the concept of the attractors and some of its properties, which is very important for the understanding of the dynamic systems theory. In the second chapter we studied topological dynamic systems and their properties. We explained limit sets, minimal sets and the term of recurrence. We introduced the concepts and properties of dynamic systems regarding to being topologically transitive, expansive or topologically mixing. Furthermore, next chapter, Dimensions, treated three different definitions of dimensions (Lebesgue covering dimension, Hausdorff dimension and upper box-counting dimension) and investigated what being ’finite dimensional’ implies in terms of embeddings into Euclidean spaces for each of these definitions. In the fourth chapter we presented the main subject of this paper, the Takens theorem and its proof. We also introduce some technical results. In the last chapter we gave an important example in the dynamics theory - Lorenz equations and Lorenz attractor (defined by those equations)

• Keywords: Takensov teorem; dinamički sustavi; topološki dinamički sustavi; Lebesgueova pokrivajuća dimenzija; Hausdorffova dimenzija; box dimenzija; konačnodimenzionalnost; Lorenzove jednadžbe; Lorenzov atraktor
• Keywords (english): Takens theorem; dynamic systems; topological dynamic systems; Lebesgue covering dimension; Hausdorff dimension; upper box-counting dimension; ’finite dimensional’; Lorenz equations; Lorenz attractor
• Language: croatian
• URN:NBN: urn:nbn:hr:217:229624
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-02-20 15:10:02