Qualitative Theory of Differential Equations

Code:

M4048

Acronym:

M4048

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2016/2017 - 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	8	Plano de Estudos do M:Matemática	1	-	6	56	162
			2

Teaching language

Suitable for English-speaking students

Objectives

To motivate and introduce the theory and classical methods associated with the qualitative study of differential equations.

Learning outcomes and competences

 The student must dominate the basic results of the theory of stability (local and global) and be able to use the various tools of qualitative theory in order to deduce dynamic properties of a given differential system or vector field. The student must also learn to model some problems and draw conclusions from their qualitative study.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Working method

Presencial

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

Linear Algebra, Real Analyis and Toplogy (elementary)

Program

Main topics:

- Resolution of linear differential equations via exponential operator.

- Qualitative theory of linear differential equations associated with hyperbolic linear applications. Structural stability.

- Fundamental Theory of ODE: Existence and uniqueness of solutions; maximal domains maximum; solutions that flee from compact sets;  flow associated to a differential equation; topological classification of solution curves.

- Gradient vector filds. Lyapunov functions

- Dynamics in two dimension: Dulac theorem; transversal sections  and Poincaré-Bendixson theorem. Theorem of Denjoy-Schwartz

- Hyperbolicity. Stability of equilibrium points. Hartman-Grobman theorem (without proof)

- Stable Manifild Theorem (without proof). Homoclinic phenomena.

- Hartman-Grobman theorem for periodic orbits hyperbolic.

- Lorenz Attractor.

- RLC circuits and Van der Pol Equation

- Another examples

Mandatory literature

000045556. ISBN: 0-12-349550-4
000052669. ISBN: 0-8493-8493-1
000015086. ISBN: 0-387-96649-8
000053002. ISBN: 0-387-97894-1
Luís Barreira e Cláudia Valls; Equações Diferenciais: Teoria Qualitativa, IST Press, 2010. ISBN: 978-972-8469-96-2
Jacob Palis e Welington de Melo; Introdução aos Sistemas Dinâmicos, Projecto Euclides - IMPA, 1978
000073410. ISBN: 0-12-349703-5

Teaching methods and learning activities

For each class: 1h30m dedicated to the presentation of the theory and  to the discussion of examples in the blackboard,  30m dedicated to the practical component (which is expected to have a strong student participation).

keywords

Physical sciences > Mathematics

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Teste	70,00
Trabalho escrito	30,00
Total:	100,00

Calculation formula of final grade

The resolution of exercises has an weight of 30%. There are two written tests, each with a weight 35%.
The sum of the three ratings should be greater  or equal to 10/20.
The three evaluation components are mandatory.

Observations

Other evaluations  will have a written component (10 points) and oral component (10 points). The approval requires a minimum of four points in each component and the sum of the ratings must be greater  or equal to 10.