Introduction to Dynamical Systems

Code:

M3029

Acronym:

M3029

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2022/2023 - 1S

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

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:M	7	Official Study Plan	3	-	3	28	81
L:MA	1	Official Study Plan	3	-	3	28	81

Teaching language

Suitable for English-speaking students

Objectives

The syllabus presents examples and results mainly on low dimensional dynamics in order to emphasize the geometrical, topological and probabilistic methods in this reasearch area and their broad applications in other sciences.

Learning outcomes and competences

The syllabus is an introduction to a recent but outstanding research mathematical area. The list of topics chosen for this course includes a preliminary classical content and results from recent research, besides several examples and applications.

Working method

Presencial

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

Analysis, Measure Theory and Topology.

Program

1. Discrete dynamical systems. Basic notions: orbit, periodicity, recurrence, wandering points, transitivity, genericity, conjugacy, stability, minimality.

2. Detailed analysis of several examples of the interval and the plane

3. Homeomorphisms of the unit circle: Theorems of Poincaré and Denjoy.

4. Sharkowskii order (period 3 implies chaos).

6. Dynamical proofs of some results in Number Theory.

Mandatory literature

Hirsch Morris W.; Differential equations, dynamical systems, and introduction to chaos. ISBN: 0-12-349703-5
Melo Welington de; One-dimensional dynamics. ISBN: 3-540-56412-8
Shub Michael; Global stability of dynamical systems. ISBN: 0-387-96295-6
Robinson Clark; Dynamical systems. ISBN: 0-8493-8493-1

Teaching methods and learning activities

The content of the syllabus is presented in the lectures, where examples, exercises and problems are discussed. Reading suggestions, additional bibliography and other resources are available for students at the FCUP library.

 

keywords

Physical sciences > Mathematics

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	50,00
Apresentação/discussão de um trabalho científico	50,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	47,00
Frequência das aulas	28,00
Apresentação/discussão de um trabalho científico	10,00
Total:	85,00

Eligibility for exams

There will be no record of absences.

Calculation formula of final grade

The evaluation consists of a written exam E (marks between 0 and 20) and a presentation of work in class A (marks between 0 and 20). The final grade is NF=(E+A)/2.

Internship work/project

Not applicable.

Special assessment (TE, DA, ...)

Not applicable.

Classification improvement

According to the FCUP regulations, the students have access to an exam to improve their grades.

Observations

Article 13th of the General Regulation for Students’ Evaluation in the University of Porto, approved the 19th May 2010: "Any student who commits fraud during an exam or test fails that exam and will face disciplinary charges by the University."