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 (ODE), 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. (4 hours exposition approximately)
Basic bibliography:
Bibliografía básica:
AGARWAL, R. P.; O’REGAN, D.; An Introduction to Ordinary Differential Equations. Springer, 2008.
DIZ PITA, É. Sistemas de ecuacións diferenciais autónomos en R^n. Unidades Didacticas USC. Editorial: Servicio de Publicacións e Intercambio Científico USC, 2022
https://dx.doi.org/10.15304/9788418445965 ISSN/ISBN 9788418445965
DIZ PITA, É.; OTERO ESPINAR, M. V.; Estabilidade de sistemas lineares. Unidades Didacticas USC. Editorial: Servicio de Publicacións e Intercambio Científico USC, 2022
https://dx.doi.org/10.15304/9788419155719, ISSN/ISBN 978-84-19155-71-9
DIZ PITA, É.; OTERO ESPINAR, M. V.; Estabilidade de sistemas non lineares. Unidades Didacticas USC. Servicio de Publicacións e Intercambio Científico USC, 2022
https://dx.doi.org/10.15304/9788419155702, ISSN/ISBN 978-84-19155-70-2
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.
OTERO ESPINAR, M. V.; DIZ PITA, É.; Ecuacións diferenciais ordinarias: modelos esenciais aplicados a distintos ámbitos científicos. Esenciais USC. Servicio de Publicacións e Intercambio Científico USC, 2021. ISSN/ISBN 9788418445781
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.
It will be followed the general methodological indications established in the Title of degree of Mathematics of the USC.
Teaching is scheduled in theoretical and interactive classes.
The theoretical classes will be devoted to the presentation and development of the essential contents of the subject.
The interactive classes will be devoted to the presentation of examples and problem solving (combining both theory and applications).
It will be promoted the maximum participation of students on the various classes of interactive teaching laboratory, where the discussion and debate with students on aspects of the subject and the resolution of the proposed tasks will aim to practice and improve their knowledge, and to work to achieve some of the competences mentioned.
Tutorials will be in person or through email.
It will follow the general criterion of assessment established in the Memory of the Title of Degree in Mathematics of the USC.
For the calculation of the final qualification (FQ) we will take into account the qualification of the continuous assessment (CA) and the qualification of the final exam (FE).
The final qualification will be computed using the following formula:
FQ=Maximum{0.6*FE+0.4*CA; FE}.
The continuous assessment will consist of carrying out the following activities:
- resolution of problems, which might be indivual or in groups, during lessons, with notes (A);
- a midterm test (without notes) that will not reduce the amount of contents of the final exam (B).
This way, the continuous assessment will be computed with the following formula:
AC=0.5*A+0.5*B.
The activities of the continuous assessment will be similar in all the groups. The continuous assessment will be preserved for the second opportunity.
The final exam will consist of the resolution of theoretical and practice questions similar to the ones studied during the development of the subject. The final exam will be the same for all the groups.
Those students who do not take the final exam will be considered as not presented.
For the second opportunity, the final qualification will be computed using the same formula.
Warning: In cases of fraudulent performance of exercises or tests, the provisions of the "Normativa de avaliación do rendemento académico dos estudantes e de revisión de cualificacións" will apply.
CLASSROOM WORK
Expositive classes (14 h)
Seminar classes (14 h)
Laboratory classes (14 h)
Tutoring in very small groups (2 h)
Total work hours in the classroom: 44
PERSONAL STUDENT WORK
Self-study individual or group (42 hours)
Writing of exercises, conclusions and other works (15 h)
Programming/experimentation and other computer work (7.5 h)
Suggested readings, activities in library or similar (5h)
Total hours student's personal work: 68.5
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 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.
Lucia Lopez Somoza
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- lucia.lopez.somoza [at] usc.es
- Category
- PROFESOR/A PERMANENTE LABORAL
Jorge Rodríguez López
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- jorgerodriguez.lopez [at] usc.es
- Category
- PROFESOR/A PERMANENTE LABORAL
| Monday | |||
|---|---|---|---|
| 13:00-14:00 | Grupo /CLIL_06 | Galician | Computer room 3 |
| Tuesday | |||
| 09:00-10:00 | Grupo /CLIL_04 | Galician | Computer room 2 |
| 10:00-11:00 | Grupo /CLIL_05 | Galician | Computer room 2 |
| 12:00-13:00 | Grupo /CLE_02 | Galician | Classroom 06 |
| 13:00-14:00 | Grupo /CLE_01 | Galician | Classroom 02 |
| Thursday | |||
| 09:00-10:00 | Grupo /CLIS_04 | Galician | Classroom 07 |
| 10:00-11:00 | Grupo /CLIS_03 | Galician | Classroom 06 |
| 11:00-12:00 | Grupo /CLIL_02 | Galician | Computer room 3 |
| 12:00-13:00 | Grupo /CLIL_03 | Galician | Computer room 3 |
| 13:00-14:00 | Grupo /CLIL_01 | Galician | Computer room 3 |
| Friday | |||
| 10:00-11:00 | Grupo /CLIS_01 | Galician | Classroom 02 |
| 11:00-12:00 | Grupo /CLIS_02 | Galician | Classroom 06 |
| 01.19.2026 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |
| 06.15.2026 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |