ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 102 Hours of tutorials: 6 Expository Class: 18 Interactive Classroom: 24 Total: 150
Use languages Spanish, Galician
Type: Ordinary subject Master’s Degree RD 1393/2007 - 822/2021
Departments: Applied Mathematics
Areas: Applied Mathematics
Center Faculty of Mathematics
Call: First Semester
Teaching: With teaching
Enrolment: Enrollable | 1st year (Yes)
I. NUMERICAL METHODS FOR INITIAL VALUE PROBLEMS (IVPs) ASSOCIATED WITH ORDINARY DIFFERENTIAL EQUATIONS (ODEs):
1. To know the most common methods for the numerical resolution of IVPs for ODEs.
2. To become familiar with the concepts of convergence and order, related to accuracy, and with numerical stability, related to error blowup.
3. To observe the phenomena mentioned in the previous point, as well as the effect of rounding errors on convergence, by implementing some of the studied methods on a computer.
II. DYNAMICAL SYSTEMS:
1. To proficiently handle some analytical methods of integrating ordinary differential equations.
2. To understand and be able to analyze low-dimensional dynamical systems.
3. To understand the basic concepts of bifurcations and know how to apply them to specific problems.
4. To use dynamical systems to model and analyze problems of industrial interest.
I. NUMERICAL METHODS FOR INITIAL VALUE PROBLEMS (IVPs) ASSOCIATED WITH ORDINARY DIFFERENTIAL EQUATIONS (ODEs):
1. Preliminaries: Basic formulas for numerical differentiation and integration, contraction mapping theorem, solving nonlinear systems of equations using fixed-point and Newton’s methods, epsilon-type and delta-type stopping criteria.
2. Concept of an initial value problem (IVP) for ODEs. Existence and uniqueness theorem for IVP solutions. Idea of numerical solutions for IVPs.
3. Description and interpretation of Euler methods: Explicit and implicit.
4. Family of theta-methods. Trapezoidal rule.
5. Convergence. Order of convergence. Consistency and stability.
6. Influence of round-off errors.
7. MATLAB® commands for solving IVPs.
8. Example of a stiff problem. Numerical stability.
9. High-order methods:
9.a. Single-step nonlinear methods: Runge-Kutta (RK) methods.
9.b. Linear multistep methods (LMMs):
9.b.i. Concept of LMMs. Starting procedure. Order theorem.
9.b.ii. LMMs based on numerical integration:
• Adams-Bashforth methods.
• Adams-Moulton methods.
• Nyström methods.
• Milne-Simpson methods.
9.b.iii. LMMs based on numerical differentiation: BDF methods.
II. DYNAMICAL SYSTEMS:
1. Basic concepts: differentiable parametrized curve, eigenvalue, eigenvector, algebraic multiplicity, geometric multiplicity.
2. Linear dynamical systems in R^n:
2.a. Decoupled systems, phase portrait, equilibrium point, stable subspace, unstable subspace.
2.b. Systems with a diagonalizable matrix.
2.c. Systems with a general matrix (not necessarily diagonalizable), fundamental theorem for linear systems, matrix exponential, solving linear systems, higher-order linear systems.
2.d. Linear systems in R^2:
2.d.i. Classification of the origin for systems with a regular matrix: saddles, nodes (stable and unstable), foci (stable and unstable), centers; sinks and sources.
2.d.ii. Examples with a singular matrix, degenerate equilibrium points.
2.e. Characterization of sinks and sources.
3. Nonlinear dynamical systems in R^n:
3.a. Autonomous systems.
3.b. Fundamental existence and uniqueness theorem.
3.c. Linearization.
3.d. Hyperbolic equilibrium points. Hartman-Grobman theorem.
3.e. Classification of hyperbolic equilibrium points: sinks, sources, and saddles.
3.f. Stable, asymptotically stable, and unstable equilibrium points.
3.g. Lyapunov functions method.
3.h. Concept of bifurcation; concept of chaos.
I. NUMERICAL METHODS FOR INITIAL VALUE PROBLEMS (IVPs) ASSOCIATED WITH ORDINARY DIFFERENTIAL EQUATIONS (ODES):
BASIC BIBLIOGRAPHY:
1. Ascher, U. M., and Petzold, L. R. (1998) Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Philadelphia, PA: SIAM.
2. Hairer, E., Nørsett, S. P., and Wanner, G. (1993) Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd revised ed. Berlin: Springer. (First edition: 1987). Available online.
3. Isaacson, E., and Keller, H.B. (1994) Analysis of Numerical Methods. New York, NY: Dover Publications. (Reprint of the 1966 edition published by Wiley).
4. Iserles, A. (2008) A First Course in the Numerical Analysis of Differential Equations, 2nd ed. Cambridge: Cambridge University Press. (First published in 1997). Available online.
5. Lambert, J. D. (1991) Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. Chichester, UK: Wiley.
6. Stoer, J., and Bulirsch, R. (2002) Introduction to Numerical Analysis, 3rd ed. New York, NY: Springer. (First edition: 1980). Available online.
COMPLEMENTARY BIBLIOGRAPHY:
1. Butcher, J. Ch. (2016) Numerical Methods for Ordinary Differential Equations, 3rd ed. Chichester, UK: Wiley. (First edition: 2003). Available online.
2. Crouzeix, M., and Mignot, A. L. (1989) Analyse Numérique des Équations Différentielles, 2nd ed. Paris: Masson. (First edition: 1984).
3. Dekker, K., and Verwer, J. G. (1984) Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. Amsterdam: Elsevier Science Publishers B. V.
4. Hairer, E., and Wanner, G. (1996) Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, 2nd ed. Berlin: Springer. (First edition: 1991). Available online.
5. Henrici, P. (1962) Discrete Variable Methods in Ordinary Differential Equations. New York, NY: Wiley.
6. Kincaid, D. R., and Cheney, E. W. (1991) Numerical Analysis: Mathematics of Scientific Computing. Pacific Grove, CA: Brooks/Cole.
7. Lambert, J. D. (1988) Computational Methods in Ordinary Differential Equations. London: Wiley. (Reprint of the 1st ed. 1973).
8. Quarteroni, A., Sacco, R., and Saleri, F. (2007) Numerical Mathematics, 2nd ed. New York, NY: Springer. (First edition: 2000). Available online.
II. DYNAMICAL SYSTEMS:
BASIC BIBLIOGRAPHY:
1. Perko, L. (2000) Differential Equations and Dynamical Systems, 3rd ed. Texts in Applied Mathematics, vol. 7. New York, NY: Springer. Available online.
2. Hirsch, M. W., and Smale, S. (1974) Differential Equations, Dynamical Systems, and Linear Algebra. Pure and Applied Mathematics. New York, NY: Academic Press. Available online.
COMPLEMENTARY BIBLIOGRAPHY:
1. Guckenheimer, J., and Holmes, P. (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York, NY: Springer. Available online.
2. Hale, J. K., and Koçak, H. (1991) Dynamics and Bifurcations. New York, NY: Springer. Available online.
3. Hairer, E., Nørsett, S. P., and Wanner, G. (1993) Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd revised ed. Berlin: Springer. (First edition: 1987). Available online.
Basic and General Competencies:
CG1: To possess knowledge that provides a foundation or opportunity for originality in the development and/or application of ideas, often in a research context, and to translate industrial needs into R&D&i projects in the field of Industrial Mathematics.
CG4: To communicate conclusions—along with the underlying knowledge and rationale—clearly and unambiguously to both specialized and non-specialized audiences.
CG5: To possess the learning skills required to continue studying autonomously and to undertake doctoral studies successfully.
Specific Competencies:
CE3: To assess whether a process model is mathematically well-posed and physically well-formulated.
Specialization in “Modeling”:
CM1: To derive both qualitative and quantitative information from models using analytical techniques.
The competencies listed above, as well as those described on page 8 of
https://assets.usc.gal/sites/default/files/plan/2021-09/Matema%CC%81tic…,
are developed in class and assessed according to the evaluation criteria outlined in the "Assessment System" section.
1. Planning the content of each class.
2. Explanation on the board (lecture) or equivalent using videoconferencing.
3. Programming some methods on the computer.
EVALUATION CRITERIA FOR THE FIRST ATTEMPT:
The competencies CG1, CG4, CG5, as well as CE3 and CM1, will be assessed through the following process:
To pass the course, students must submit all assigned exercises and programming assignments by the deadlines set by the instructors. The final grade will be determined by a written exam in which:
• Each of the two course components (Numerical Methods for ODEs and Dynamical Systems) will account for 50% of the final grade.
• In the Numerical Methods for ODEs examination, 30% of the grade will be based on questions related to programming assignments.
The course examinations for both components - Numerical Methods for ODEs and Dynamical Systems - are administered on separate dates. The first examination is conducted upon completion of the initial instructional unit, typically scheduled for early November, while the second examination takes place during the standard end-of-semester assessment period.
The end-of-semester examination evaluates only the component not assessed in the November exam. Importantly, the first-examined component is not retested on this end-of-semester examination date.
Class attendance (or lack thereof) will not affect the final grade.
EVALUATION CRITERIA FOR THE SECOND ATTEMPT:
The same criteria as for the first evaluation attempt will apply. However, in this second assessment opportunity, the written examinations for both course components - Numerical Methods for ODEs and Dynamical Systems - will be administered on the same date.
Partial grades from the first examination attempt are not transferable to the second attempt. Students who pass one component examination but fail the overall course must retake both examinations during the second evaluation period.
Personal work hours, including class hours: approximately 150 hours (25 hours per ECTS).
Programming assignments will be conducted using MATLAB®.
The instructional sequence of course components - Numerical Methods for ODEs and Dynamical Systems - will be announced at the beginning of each academic year.
Cases involving fraudulent completion of exercises or examinations will be processed according to the USC's “Normativa de avaliación do rendemento académico dos estudantes e de revisión de cualificacións” (“Regulations for the Assessment of Student Academic Performance and Grade Review”).
Óscar López Pouso
Coordinador/a- Department
- Applied Mathematics
- Area
- Applied Mathematics
- Phone
- 881813228
- oscar.lopez [at] usc.es
- Category
- Professor: University Lecturer
| Monday | |||
|---|---|---|---|
| 09:00-10:00 | Grupo /CLE_01 | Spanish | Computer room 5 |
| Thursday | |||
| 11:00-12:00 | Grupo /CLE_01 | Spanish | Computer room 5 |
| Friday | |||
| 10:00-11:00 | Grupo /CLE_01 | Spanish | Computer room 5 |
| 11.06.2025 16:00-20:00 | Grupo /CLE_01 | Computer room 5 |
| 12.18.2025 16:00-20:00 | Grupo /CLE_01 | Computer room 5 |
| 06.09.2026 10:00-14:00 | Grupo /CLE_01 | Computer room 5 |