Qualitative Theory of Differential Equations

Code:

M4132

Acronym:

M4132

Level:

400

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2025/2026 - 1S

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	17	Official study plan since 2024/2025	1	-	6	42	162
			2

Teaching language

English

Objectives

Learning outcomes and competences

Working method

Presencial

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

Basic courses of Linear Algebra, Real Analysis and Analysis in several variables.

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 ordinary differential equations: 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 fields. Lyapunov functions.
  - Dynamics in two dimension: transversal sections and Poincaré- Bendixson theorem. 
   - Hyperbolicity. Stability of equilibrium points. Hartman-Grobman theorem (without proof). Stable Manifold Theorem (without proof). 

Other topics to be addressed in case time permits:   
  - Homoclinic/Heteroclinic  phenomena.
  - Hartman-Grobman theorem for periodic orbits hyperbolic.
  - Lorenz attractor.
  - Van der Pol Equation. 

Mandatory literature

Morris W. Hirsch; Differential equations, dynamical systems, and introduction to chaos. ISBN: 0-12-349703-5
Morris W. Hirsch; Differential equations, dynamical systems, and linear algebra. ISBN: 0-12-349550

Complementary Bibliography

Martin Braun; Differential equations and their applications. ISBN: 0-387-90266-X
J. Espinar, M. Viana; Differential equations: a dynamical systems approach to theory and practice, American Mathematical Society, Graduate Studies in Mathematics vol. 212, 2021.

Teaching methods and learning activities

Theoretical lectures for introducing the concepts.
Exercise Sessions where student participation will be encouraged.

Evaluation Type

Distributed evaluation without final exam

Assessment Components

designation	Weight (%)
Teste	100,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	120,00
Frequência das aulas	42,00
Total:	162,00

Eligibility for exams

The absence from lectures will not be taken into account.

Calculation formula of final grade

There will be two tests during the semester, both worth 10 points. The final grade will be the sum of the grades of the two tests.

Any student can choose not to submit to continuous evaluation and obtain the final classification performing the examination in the second examination period (Época de Recurso).

In any case, a student with a final grade ≥ 16.5 may eventually have to make a presentation (oral or written) in a topic arranged between the student and the course teacher.

All registered students are admitted, without restrictions, to the tests and exams.

Special assessment (TE, DA, ...)

According to the General Evaluation Rules.

Any student asking for an exam because of special conditions of his registration will do a written exam, but possibly, only, after an extra written or oral examination, in order to check if the student has a minimum knowledge about the unit so that he can do the special exam.

Classification improvement

The general evaluation rules apply.