Hyperbolic Dynamics

Code:

M518

Acronym:

M518

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2021/2022 - 2S

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

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
IUD-M	3	PE do Prog Inter-Univ Dout Mat	1	-	9	60	243

Teaching language

Portuguese

Objectives

Study local and global features of uniformly hyperbolic dynamical systems, focusing on its stability and classification. Classification of such dynamical systems according to the geometric and topological features of its basic pieces.

Learning outcomes and competences

Identify hyperbolicity in discrete-time and continuous-time dynamical systems. Describe the complexity of basic sets in dynamical systems. Identify robustness and stability in dynamical systems.

Working method

Presencial

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

Basic tools from Analysis and Geometry

Program

1 - Basic concepts in dynamical systems 2 - Basic results in one dimensional dynamics 3 - Linear systems 4 - Hyperbolic sets: definitions and robustness 5- Shadowing lemma 6 - Stability of hyperbolic dynamical systems 7 -Spectral decomposition 8- Beyond uniform hyperbolicity

Mandatory literature

Jacob Palis Jr.; Geometric theory of dynamical systems. ISBN: 0-387-90668-1
Michael Shub; Global stability of dynamical systems. ISBN: 0-387-96295-6
M. C. Irwin; Smooth dynamical systems. ISBN: 0-12-374450-4
Michael Brin; Introduction to dynamical systems. ISBN: 0-521-80841-3

Complementary Bibliography

Welington de Melo; One-dimensional dynamics. ISBN: 3-540-56412-8
Anatole Katok; Introduction to the modern theory of dynamical systems. ISBN: 0-521-57557-5
Clark Robinson; Dynamical systems. ISBN: 0-8493-8493-1
Morris W. Hirsch; Differential equations, dynamical systems, and linear algebra. ISBN: 0-12-349550
Z. Nitecki; Global theory of dynamical systems. evanston. 1979. ISBN: 3-540-10236-1
Yakov B. Pesin; Dimension theory in dynamical systems. ISBN: 0-226-66221-7
Sergei Yu. Pilyugin; Shadowing in dynamical systems. ISBN: 3-540-66299-5

Teaching methods and learning activities

Classes in the blackboard and autonomous study

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)
Apresentação/discussão de um trabalho científico	30,00
Total:	30,00

Eligibility for exams

Active participation in the classes

Calculation formula of final grade

Each student can opt by either:

1. Final exam (0-20)

2. Seminar (0-10) and a shorter test (0-10)