Introduction to Dynamical Systems

Code:

M312

Acronym:

M312

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2014/2015 - 2S

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:AST	1	Plano de Estudos a partir de 2008	3	-	7,5	63	202,5
L:B	0	Plano de estudos a partir de 2008	3	-	7,5	63	202,5
L:F	0	Plano de estudos a partir de 2008	3	-	7,5	63	202,5
L:G	0	P.E - estudantes com 1ª matricula anterior a 09/10	3	-	7,5	63	202,5
		P.E - estudantes com 1ª matricula em 09/10	3	-	7,5	63	202,5
L:M	5	Plano de estudos a partir de 2009	1	-	7,5	63	202,5
			2
			3
L:Q	0	Plano de estudos Oficial	3	-	7,5	63	202,5

Teaching language

Portuguese

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

Notions: orbit, periodicity, recurrence, wandering points, transitivity, genericity, conjugacy, stability. Homeomorphisms of the real line and the circle. Theorems of Poincaré, Denjoy and Peixoto.
Shifts. Smale's horseshoe. Markov partitions and codification. Linear dynamical systems. Anosov diffeomorphisms.
Measures invariant by a dynamics. Ergodicity, mixing and minimality. Poincaré Recurrence Theorem. Birkhoff Ergodic Theorem. Kac Theorem. 
Existence of probability measures invariant by continuous dynamics on compact metric spaces. Examples and applications.

Mandatory literature

Palis Jr. Jacob; Geometric theory of dynamical systems. ISBN: 0-387-90668-1
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
Parthasarathy K. R.; Probability measures on metric spaces
Halmos Paul R.; Measure theory. ISBN: 0-387-90088-8
Robinson Clark; Dynamical systems. ISBN: 0-8493-8493-1
Nitecki Zbigniew; Differentiable dynamics. ISBN: 0-262-64011-2
Walters Peter; An introduction to ergodic theory. ISBN: 0-387-95152-0

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 > Chaos theory
Physical sciences > Mathematics > Probability theory

Evaluation Type

Evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	50,00
Prova oral	50,00
Total:	100,00

Eligibility for exams

There will be no record of absences.

Calculation formula of final grade

According to the regulations of UP and FCUP, the teacher and the students agreed that the evaluation consists of either an exam (0-10) and a seminar (0-10) or just an exam (0-20).

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

Jury: Maria Pires de Carvalho, Fernando Jorge Moreira