Dynamical Systems

Code:

M4047

Acronym:

M4047

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2021/2022 - 2S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Master in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
M:M	13	Plano Oficial do ano letivo 2021	1	-	6	56	162

Teaching language

Suitable for English-speaking students

Objectives

Since this is a first course in Dynamical Systems one aims to introduce the basic concepts and some classical results,  based on examples but also including rigorous proofs.

Learning outcomes and competences

The student must dominate the basic results of the theory of low dimensional dynamical systems (local and global) and be able to use the various tools of in order to deduce dynamic properties of a given dynamical system.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Linear Algebra, Analysis and elementary Topology.

Program

1) Basic concepts
2) Unidimensional dynamics:
- Sharkovski Theorem
- the quadratic family
- one-side shift
- hyperbolicity
- homeomorphisms of the circle
- topological entropy
3) Dynamics in dimensiom two:
- Linear Anosov on the torus
- Smale's horseshoe
- hiperbolicity.
4) Additional topics (essentially without proofs):
Hartman-Grobman's Theorem, Stable manifold theorem,  local and global stability.

Mandatory literature

Robert L. Devaney; An introduction to Chaotic Dynamical Systems (second edition), Addison-Wesley Publishing Company, 1989. ISBN: 0-201-13046-7
Clark Robinson; Dynamical Systems, Stability, Symbolic Dynamics, and Chaos, CRC Press, 1999. ISBN: 0-8493-8495-8
Luís Barreira e Caudia Valls; Sistemas Dinâmicos: uma introdução, ISTPress, 2012. ISBN: 978-989-841-15-3

Teaching methods and learning activities

For each class: 1h50m dedicated to the presentation of the theory,  and  to the discussion of examples in the blackboard,  being 30m dedicated to the practical component.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	60,00
Teste	40,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	50,00
Frequência das aulas	50,00
Total:	100,00

Eligibility for exams

not applicable

Calculation formula of final grade

Sum equal or bigger than 9,5 (in 20) in the two written examinations