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Unidimensional dynamical systems: lexicon, elemental concepts and some important theorems

Grant number: 16/21492-7
Support Opportunities:Scholarships in Brazil - Scientific Initiation
Effective date (Start): March 01, 2017
Effective date (End): February 28, 2018
Field of knowledge:Physical Sciences and Mathematics - Mathematics
Principal Investigator:Liane Bordignon
Grantee:Gabriel Longatto Clemente
Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil


The aim of this project is to give the student an elemental background in one dimensional discrete dynamics systems, preparing him to research in this subject. It proposes the study of the first chapter of "An introduction to chaotic dynamical systems" of R. Devaney, complemented by more advanced texts addressed to researchers in the area, improving the scientific maturity of the student. The candidate will study fundamental concepts of dynamical systems, including symbolic dynamics, Devaney's chaos definition, Sarkovskii's Theorem, structural stability, dynamics in $S^1$ and fragments of kneading theory. At the end, the student will have seen a description of the very illustrative behaviour of the family $F_\mu(x)= \mu x(1-x)$, $\mu \in \mathbb{R}, \mu \geq 1$, including the period duplication phenomenon and the Poincaré Classification Theorem for $S^1$ homeomorphisms. (AU)

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