Qualitative Theory of Differential Equations

Code:

M4132

Acronym:

M4132

Level:

400

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2023/2024 - 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	12	Plano Oficial do ano letivo 2021	1	-	9	72	243
			2

Teaching language

Portuguese and english

Objectives

Learning outcomes and competences

Working method

Presencial

Program

 - 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 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). 

   - Homoclinic/Heteroclinic  phenomena.
  - Hartman-Grobman theorem for periodic orbits hyperbolic.
  - RLC circuits and Van der Pol Equation. 

Mandatory literature

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

Teaching methods and learning activities

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

Evaluation Type

Distributed evaluation with final exam

Assessment Components

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

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	171,00
Frequência das aulas	72,00
Total:	243,00

Eligibility for exams

The absence from lectures will not be taken into account.

Calculation formula of final grade

The final exam will correspond to 70% of the average while the test to be applied in the middle of the course will correspond to the remaining 30%.