ECTS credits ECTS credits: 3
ECTS Hours Rules/Memories Student's work ECTS: 51 Hours of tutorials: 3 Expository Class: 9 Interactive Classroom: 12 Total: 75
Use languages Spanish, Galician
Type: Ordinary subject Master’s Degree RD 1393/2007 - 822/2021
Departments: Statistics, Mathematical Analysis and Optimisation
Areas: Mathematical Analysis
Center Faculty of Mathematics
Call: Second Semester
Teaching: With teaching
Enrolment: Enrollable | 1st year (Yes)
The terminology and basic concepts of the theory of the dynamical systems are introduced, as well as elementary topics on the behaviour of the differential systems: in R^n the theorems of the invariant manifolds and Hartman-Grobman are studied and, for the particular case of the plane, the tools for the study of some non elementary singularities (in particular, non degenerated singularities).
As examples of discrete dynamical systems, the quadratic application or the horseshoe of Smale are studied.
1.- Generalities: the general concept of dynamical system. Orbits and limit sets. (3 lectures)
2.- Examples of dynamical systems: flows and discrete dynamical systems. (4 lectures)
3.- Equivalence and conjugation. The structural stability. (3 lectures)
4.- Recurrence. (1 lecture)
5.- Dynamical systems in R^n. Local study: the Hartman-Grobman and the invariant manifolds theorems. (4 lectures)
6.- Dynamical systems in the plane. Study of the critical points. (4 lectures)
Basic bibliography
Devaney, R.L.: An introduction to chaotic dynamical systems. Benjamin C. (1986) (catalog number: 1202 341; 58 205)
Dumortier, F., Llibre, J., Artés, J.C.: Qualitative Theory of Planar Differential Systems. UniversiText, Springer-Verlag, (2006) URL https://link.springer.com/book/10.1007/978-3-540-32902-2
Perko, L.: Differential Equations and Dynamical Systems. Springer (1991) (catalog number: 34 400 A) URL https://link.springer.com/book/10.1007/978-1-4684-0249-0
Complementary bibliography
Guckenheimer, J., Holmes, P.: Nonlinear oscilations, dynamical systems, and bifurcations of vector fields. Springer-Verlag (1983) URL https://link.springer.com/book/10.1007/978-1-4612-1140-2
Hubbard, J.H., West, B.H.: Differential Equations: A Dynamical Systems Approach. Texts in Applied Mathematics, 18. Springer-Verlag (1995) URL https://link.springer.com/book/10.1007/978-1-4612-0937-9
Irwin, M.C.: Smooth Dynamical Systems. Academic Press (1980) (catalog number: 34 347)
Palis, J., de Melo, W.: Geometric Theory of Dynamical Systems. Springer (1982) (catalog number: 37 35 A) URL https://link.springer.com/book/10.1007/978-1-4612-5703-5
Sotomayor, J.: Liçoes de equaçoes diferenciais ordinarias. IMPA (1979) (catalog number: 1202 83; 34 165)
Yan-Qian, Y.: Theory of Limit Cycles. Translations of Mathematical Monographs, 66. AMS (1986) (catalog number: 34 101)
• Acquisition of high-level mathematical tools for several applications, meeting the expectations of graduates in mathematics and other basic sciences (CG02).
• To know the gret influence of Dynamical System in diverse fields of contemporary mathematics (CG03).
• To train for problem analysis and problem solving in new or unfamiliar environments within broader contexts (CG04).
• To prepare for decision-making from abstract considerations in order to organize, plan and resolve complex issues (CG05).
• To train for the study and research in developing mathematical theories (CE01).
• To apply mathematical tools in various fields of science, technology and social sciences (CE02).
• To develop the skills required for oral and written transmission of mathematical knowledge with formal correctness and communication efficiency (CE03).
• To make use of bibliography and of bibliography search tools, including Internet usage (CT01).
• To manage time and other available resources in an optimal way and to maximize the ability to work in collaborative environments (CT02).
The general directions of the master will be followed.
The methodology used will be based on: presentations by the teacher, completion and presentation by the students concerning some works proposed, development of exercises, discussions in the classroom and complementary reading.
The listed skills will be exercised with the proposed training activities.
There will be a virtual course available to students.
The expository and interactive teaching will be face-to-face.
The tutorials will be mainly in person and they might be partially virtual through the email or MS Teams.
The general directions of the master will be followed.
The evaluación will be carried out through continuous assessment, based on the completion of tasks, delivery of works, or presentations at the classroom. However, after the publication of the results of the continuous assessment, a voluntary final test will be proposed to give the students the opportunity to improve the qualification. If the student obtains a continuous assessment less than 5 points, then the final test is compulsory and the final qualification is the maximum between the continuous assessment mark and the final test qualification.
Through the different activities proposed, the acquisition of the mentioned skills will be evaluated.
The continuous evaluation and the final test, if necessary, will be done in person.
In the second opportunity, the continuous assessment obtained during the course will be taken into account, and a compulsory presential final test will be proposed. The qualification will be the maximum between the continuous assessment mark and the final test qualification.
Warning: In cases of fraudulent performance of exercises or tests (plagiarism or improper use of technologies), the provisions of the “Normativa de avaliación do rendemento académico dos estudantes e de revisión de cualificacións” will be applied.
ON-SITE WORK AT CLASSROOM
Blackboard classes (18 h)
Laboratory interactive classes (6 h)
Tutorials in very small groups or individualized (3 h)
Total hours on-site work at classroom: 27.
PERSONAL WORK OF THE STUDENT
Autonomous individual study or in group (30 h)
Writing exercises, conclusions and other works (15 h)
Programming / experimentation and other works at computer (3h)
Total hours personal work of the student: 48.
Basic knowledge of differential equations.
Rosana Rodríguez López
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813368
- rosana.rodriguez.lopez [at] usc.es
- Category
- Professor: University Lecturer
Lucia Lopez Somoza
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- lucia.lopez.somoza [at] usc.es
- Category
- Professor: LOU (Organic Law for Universities) PhD Assistant Professor
| Wednesday | |||
|---|---|---|---|
| 12:00-13:00 | Grupo /CLE_01 | Galician | Classroom 10 |
| Thursday | |||
| 12:00-13:00 | Grupo /CLIL_01 | Galician | Classroom 10 |
| 05.30.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 10 |
| 06.21.2026 10:00-14:00 | Grupo /CLE_01 | Classroom 10 |