ECTS credits ECTS credits: 4.5
ECTS Hours Rules/Memories Student's work ECTS: 74.2 Hours of tutorials: 2.25 Expository Class: 18 Interactive Classroom: 18 Total: 112.45
Use languages Spanish, Galician
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Departments: Statistics, Mathematical Analysis and Optimisation
Areas: Mathematical Analysis
Center Faculty of Mathematics
Call: First Semester
Teaching: With teaching
Enrolment: Enrollable
Introduce students to the field of qualitative theory of Ordinary Differential Equations, both from a theoretical and practical point of view.
To show the interest of the qualitative study of the ODE since it allows to deduce the behavior of the solutions, without needing to solve the studied system. This resolution can be very complicate in linear systems of high dimension, being impossible in most non-linear systems.
Address the stability and instability of the critical points of the system, characterizing them in the linear case and in some non-linear systems. Obtain sufficient conditions that guarantee stability and instability in the rest of the non-linear systems.
Motivate the study with mathematical models that arise from different problems in different scientific fields
Represent the evolution of the orbits of the different systems using different computer programs.
1.- Autonomous systems in R^n. Properties. Motivations and examples. Phase portrait associated with a vector field. (4 hours exposition approximately)
2.- Stability and asymptotic stability for autonomous linear systems in R^n. Applications. (3 hours exposition approximately)
3.- Stability and asymptotic stability of solutions of non-linear autonomous systems. Application to problems of physics, biology or medicine:
3.1.- Method of the first approach. (3 hours exposition approximately)
3.2.- Lyapunov's method. Region of attraction. (3 hours exposition approximately)
Basic bibliography:
AGARWAL, R. P.; O’REGAN, D.; An Introduction to Ordinary Differential Equations. Springer, 2008.
FERNANDEZ PÉREZ, C.; Ecuaciones Diferenciales I. Pirámide, 1992.
FERNANDEZ PÉREZ, C.; VEGAS MONTANER, J. M.; Ecuaciones Diferenciales II. Pirámide, 1996.
LOGEMANN, H.; RYAN, E.; Ordinary Differential Equations. Analysis, Qualitative Theory and Control. Springer, 2014.
PRECUP, R.; Ordinary Differential Equations. De Gruyter, 2018.
Complementary bibliography:
BRAUER, F.; NOHEL, J. A.; Qualitative Theory of Ordinary Differential Equations. Benjamin, 1969.
BRAUM, M.; Ecuaciones Diferenciales y sus Aplicaciones, Grupo Editorial Iberoamérica, 1990.
CODDINGTON, E. A.; LEVINSON, N.; Theory of Ordinary Differential Equations. McGraw-Hill, 1955.
CRONIN, J.; Ordinary Differential Equations. Introduction and Qualitative Theory. Chapman & Hall, 2008.
EDWARDWS, C. H.; PENNEY, D. E.; Ecuaciones Diferenciales. Prentice Hall, 2001.
GODUNOV, S. K.; Ordinary Differential Equations with Constant Coefficient. American Mathematical Society, 1997.
GUZMÁN, M.; Ecuaciones Diferenciales Ordinarias. Teoría de Estabilidad y Control. Alhambra, 1987.
HIRSCH, M. W.; SMALE, S.; Ecuaciones Diferenciales, Sistemas Dinámicos y Álgebra Lineal. Alianza Universidad, 1983.
JORDAN, D.W.; SMITH, P.; Nonlinear Ordinary Differential Equations. Oxford Univ. Press, 1999.
KRANTZ, S.; Differential Equations. Theory, Technique, and Practice with Boundary Value Poblems. CRC Press, 2016.
PEREZ GARCÍA, V. M., Problemas de Ecuaciones Diferenciales. Ariel, 2001.
SIMMONS, G. F.; Ecuaciones Diferenciales. McGraw-Hill. 1993.
SOTOMAYOR, J.; Liçoes de Equaçoes Diferenciais Ordinarias. I.M.P.A., 1979.
In this subject we will try to contribute that the students attain the following competences collected in the memory of the Degree in Mathematics of the USC.
General skills
CG1 - Know the most important concepts, methods and results of the different branches of Mathematics, together with a certain historical perspective of their development
CG2 - Collect and interpret data, information and relevant results, obtain conclusions and issue reasoned reports on scientific, technological or other problems that require the use of mathematical tools.
CG3 - Apply both acquired theoretical and practical knowledge as well as the capacity for analysis and abstraction in the definition and approach of problems and in the search for their solutions both in academic and professional contexts
CG4 - Communicate, both in writing and orally, knowledge, procedures, results and ideas in Mathematics for both a specialized and non-specialized audience.
CG5 - Study and learn autonomously, with organization of time and resources, new knowledge and techniques in any scientific or technological discipline.
Specific competences
CE1 - Understand and use mathematical language
CE2 - Know rigorous proofs of some classical theorems in different areas of Mathematics.
CE3 - Devise demonstrations of mathematical results, formulate conjectures and imagine strategies to confirm or deny them.
CE4 - Identify errors in incorrect reasoning proposing demonstrations or counterexamples.
CE5 - Assimilate the definition of a new mathematical object, relate it to others already known, and be able to use it in different contexts
CE6 - Knowing how to abstract the properties and substantial facts of a problem, distinguishing them from those that are purely occasional or circumstantial
CE7 - Propose, analyze, validate and interpret models of simple real situations, using the most appropriate mathematical tools for the purposes pursued
CE8 - Plan and execute algorithms and mathematical methods to solve problems in the academic, technical, financial or social field
CE9 - Use computer applications of statistical analysis, numerical and symbolic calculation, graphical visualization, optimization and scientific software, in general, to experiment in Mathematics and solve problems
Transversal competences
CT1 - Use bibliography and search tools for general and specific bibliographical resources of Mathematics, including Internet access
CT2 - Manage optimal working time and organize available resources, establishing priorities, alternative paths and identifying logical errors in decision making
CT3 - Check or reasonably refute the arguments of other people
CT5 - Read scientific texts both in their own language and in others of relevance in the scientific field, especially English.
In addition, the training activities to be developed have the objective that students acquire the following competences and learning outcomes related to the Differential Equations module:
Understand, learn and know how to express rigorously the concepts and techniques developed in the program.
To extract qualitative information, without their explicit resolution, of the solutions of an ordinary differential equation.
Know the characterization of the stability of linear systems.
Know techniques for studying the stability of non-linear systems.
Know relevant examples of differential equations of physics and other sciences.
Translate in terms of ordinary differential equations some problems of the applied sciences (physics, chemistry, biology, medicine, etc.).
Use the studied mathematical techniques, to understand the dynamics of the obtained equations (phase portrait, stability, etc.) and interpret the results, evaluating the strengths and weaknesses of the proposed model.
We will work in a special way: The rigorous and clear expression, both oral and written, the logical reasoning and identification of errors in the procedures, the capacity of abstraction, creativity, team work, the development of the analytical capacity in solving problems and critical attitude with different solutions.
The general methodological indications established in the Memory of the Degree in Mathematics of the University of Santiago de Compostela (USC) will be followed.
Teaching is scheduled in expository and interactive classes, some of which will consist of computer practices.
Expository Teaching (13 hours): The expository classes will be dedicated to the presentation and development of the essential contents of the subject.
Seminary and laboratory teaching (26 hours): The interactive seminary classes (13 hours) will be dedicated to the presentation of examples and problem solving (both theoretical and in the field of applications). In the interactive laboratory classes (13 hours) individual or group work will be organized and problems will be proposed to be solved by the students. In these classes, the discussion and debate with the students, as well as the resolution and exposition on their part of the proposed tasks, is fundamental so that the knowledge is practiced and strengthened and some of the mentioned competences are worked on. Students will use computer packages of symbolic calculation related to the contents of the subject.
Tutorials (2 hours): The tutoring sessions are designed to stimulate student activity outside of class. These tutorials in very small groups will serve so that the students can examine their learning process at all times.
In a generic way, in the three scenarios that will be detailed below, the virtual course will be used as a mechanism to provide students with the resources necessary for the development of the subject (notes, exercise bulletins, etc.). The fundamental difference between the three scenarios will be, on the one hand, the way of attending the tutorials and, on the other hand, the type of teaching that will be given: totally face-to-face in scenario 1, semi-presential, if necessary, in scenario 2 and totally virtual on stage three.
SCENARIO 1 (adapted normality):
The expository and interactive teaching will be face-to-face and will be complemented by the virtual course of the subject, in which the students will find bibliographic materials, problem bulletins, and other didactic materials.
The tutorials will be in person or through email.
SCENARIO 2 (distance):
Partially virtual teaching, if necessary, according to the distribution organized by the Faculty of Mathematics. For this, the virtual classroom of the course would be used with bibliographic materials, problem bulletins and other didactic materials, provided by the teaching staff and, if so established, synchronous virtual classes using the MS Teams.
The tutorials will be attended by email or through MS Teams.
SCENARIO 3 (closure of the facilities):
Totally remote teaching through the virtual course of the subject and the MS Teams. For this, the virtual classroom of the course will be used with bibliographic materials, problem bulletins and other didactic materials, provided by the teaching staff and synchronous virtual classes using the MS Teams.
The tutorials will be attended by email or through MS Teams.
For the calculation of the final grade, the continuous assessment grade and the final exam grade will be taken into account.
The continuous evaluation will consist of the realization of intermediate tasks, either through the virtual campus or as delivery of works that will be specified throughout the course. Regarding the final and second chance test, the difference between the three scenarios will consist of the way it will be carried out: face-to-face in the case of scenario 1 and telematics in scenarios 2 and 3.
Through the different activities proposed, the acquisition of skills, such as CB2, CB3, CB4, CG2, CG3, CG4, CT1, CT2, CT3, CE7, CE8, CE9, will be evaluated, of course, contextualizing the subject in the 3rd grade year, together with the capacity for teamwork and autonomous learning. The qualification obtained in the continuous evaluation will be applied in both opportunities of the same academic year (first semester and July).
There will be a final written exam, which allows checking the knowledge acquired in relation to the concepts and results of the subject and the ability to apply it to specific cases, both theoretical and applied. With the written exam that will consist of theoretical or practical questions, in addition to the specific skills of the subject, the skills CB1, CB2, CB4, CB5, CG1, CG3, CG4, CE1, CE2, CE3, CE4, CE5, CE6 will be assessed .
With the mark of the continuous formative evaluation (C) and the mark of the final face-to-face test (F) the final mark in the subject (NF) will be calculated according to the following formula:
NF = max {F, 0.5 * C 0.5 * F}
NOTE. It is possible to pass the subject without taking the final test (previous formula with F = 0). It will be understood as NOT PRESENTED who at the end of the teaching period is not in a position to pass the subject without taking the final test and does not appear at said test.
In the second opportunity, the same evaluation system will be used but with the test corresponding to the second opportunity, which will be an exam of the same type as that of the first.
SCENARIO 1 (adapted normality):
The tasks of the continuous evaluation may be carried out and/or delivered in person or through the virtual classroom.
The final test will be done in person.
SCENARIO 2 (distance):
Same procedure as that described for SCENARIO 1, with the only difference that the final test will be telematic.
SCENARIO 3 (closure of the facilities):
Both the continuous evaluation and the final test will be telematic.
Warning. In cases of fraudulent performance of the tests or tests (plagiarism or improper use of technologies), that set forth in the Regulations for the evaluation of student academic performance and review of grades will apply.
CLASSROOM WORK
Expositive classes (13 h)
Seminar classes (13 h)
Laboratory classes (13 h)
Tutoring in very small groups (2 h)
Total work hours in the classroom: 41
PERSONAL STUDENT WORK
Self-study individual or group (43 hours)
Writing of exercises, conclusions and other works (16 h)
Programming/experimentation and other computer work (7.5 h)
Suggested readings, activities in library or similar (6h)
Total hours student's personal work: 71.5
TOTAL: 112,5 hours
The students must handle with ease the subjects studied in the subjects "Introduction to Mathematical Analysis", "Continuity and Differentiability of One Real Variable Functions", "Integration of One Real Variable Functions", "Differentiation of Functions of Various Real Variables" and " Introduction to Ordinary Differential Equations".
Starting from this situation, they should work regularly (daily) and rigorously. It is essential to actively participate in the learning process of the subject, attend regularly to both theoretical and practical classes in a participatory manner, especially in classes in small groups and tutorials, asking the relevant questions.
Regarding the Methodology, the following is highlighted:
SCENARIO 2 (distance):
Partially virtual teaching, if necessary, according to the distribution organized by the Faculty of Mathematics. For this, the virtual classroom of the course would be used with bibliographic materials, problem bulletins and other didactic materials, provided by the teaching staff and, if so established, synchronous virtual classes using the MS Teams.
The tutorials will be attended by email or through MS Teams.
SCENARIO 3 (closure of the facilities):
Totally remote teaching through the virtual course of the subject and the MS Teams. For this, the virtual classroom of the course will be used with bibliographic materials, problem bulletins and other didactic materials, provided by the teaching staff and synchronous virtual classes using the MS Teams.
The tutorials will be attended by email or through MS Teams.
In relation to the evaluation, it is important to mention that it will be the same in the three situations considered, although the following must be taken into account:
SCENARIO 2 (distance):
Same procedure as described for SCENARIO 1, with the only difference that the final test will be telematic.
SCENARIO 3 (closure of the facilities):
Both the continuous evaluation and the final test will be telematic.
Maria Victoria Otero Espinar
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813170
- mvictoria.otero [at] usc.es
- Category
- Professor: University Professor
Alberto Cabada Fernandez
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813206
- alberto.cabada [at] usc.gal
- Category
- Professor: University Professor
Érika Diz Pita
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813202
- erikadiz.pita [at] usc.es
- Category
- Ministry Pre-doctoral Contract
| Monday | |||
|---|---|---|---|
| 09:00-10:00 | Grupo /CLE_02 | Galician | Classroom 08 |
| 11:00-12:00 | Grupo /CLIL_02 | Galician, Spanish | Computer room 2 |
| 12:00-13:00 | Grupo /CLIL_01 | Galician, Spanish | Computer room 3 |
| 13:00-14:00 | Grupo /CLIL_03 | Spanish, Galician | Computer room 4 |
| Tuesday | |||
| 09:00-10:00 | Grupo /CLE_01 | Galician | Classroom 09 |
| 11:00-12:00 | Grupo /CLIL_05 | Galician | Computer room 3 |
| 12:00-13:00 | Grupo /CLIL_04 | Galician | Computer room 2 |
| 13:00-14:00 | Grupo /CLIL_06 | Galician | Computer room 4 |
| Wednesday | |||
| 09:00-10:00 | Grupo /CLIS_04 | Galician | Classroom 06 |
| 10:00-11:00 | Grupo /CLIS_03 | Galician | Ramón María Aller Ulloa Main Hall |
| Thursday | |||
| 11:00-12:00 | Grupo /CLIS_02 | Galician | Classroom 06 |
| 12:00-13:00 | Grupo /CLIS_01 | Galician | Classroom 03 |
| 01.21.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 02 |
| 01.21.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 03 |
| 01.21.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |
| 01.21.2021 10:00-14:00 | Grupo /CLE_01 | Ramón María Aller Ulloa Main Hall |
| 06.29.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |