Degree | Type | Year |
---|---|---|
2500149 Mathematics | OT | 4 |
You can view this information at the end of this document.
Ordinary differential equations: existence and uniqueness of solutions of the Cauchy problem.
Linear differential systems with constant coefficients.
Linear algebra: spaces and vector subspecies, diagonalization.
This course is an introduction to the modern theory of dynamic systems. The first objective is to familiarize the student with the notion of a dynamical system and the basic concepts of this theory: stability, attractor, invariant sets, alpha and omega limits, etc. The second objective is to understand how is the local behavior, in discrete and continuous dynamical systems, near an equilibrium point or a periodic orbit. This local behavior is based on the topological classification of linear systems in R^n, both those that are determined by the flow of ordinary differential equations (continuous dynamical systems) and those that come from the iteration of functions (discrete dynamical systems). Linear systems are very important because they are the first approach to more complicated systems.
The Qualitative Theory of Differential Equations began with the work of Poincaré around 1880 in relation to his work on Celestial Mechanics and seeks to know properties of solutions without having to solve equations, among other things because the resolution is not feasible. This qualitative approach, when combined with appropriate numerical methods, is, in some cases, equivalent to having the solutions of the equation. Progress will be made in the knowledge and study, introduced previously on the plane, of the qualitative theory of differential equations in spaces of higher dimension. Emphasizing the local structure of equilibrium points (degenerate and non-degenerate) and the stability of their periodic orbits.
Finally, we introduce the techniques to study discrete global dynamics. The main example will be the unimodal maps. They (for some parameter values) present a dynamic that simply leads to the notion of a chaotic system. For these systems, the numerical approach is not feasible, and to understand its dynamics new tools are needed. Chaotic systems are often presented in applications (problems of weather forecasting, electrical circuits, etc.).
1. Dynamical systems in Euclidean spaces.
2. Study of local dynamics, discrete and continuous in Rn.
3. Global dynamics in continuous systems.
4. Global dynamics in discrete systems.
Title | Hours | ECTS | Learning Outcomes |
---|---|---|---|
Type: Directed | |||
Problem solving classes | 14 | 0.56 | |
Seminars | 6 | 0.24 | |
Theoretical lessons | 29 | 1.16 | |
Type: Autonomous | |||
Exam Preparation | 15 | 0.6 | |
Problem solving | 42 | 1.68 | |
Study of the theoretical part | 32 | 1.28 |
The subject has, during the semester and per week, two hours of theoretical lessons and one hour more to help to solve the typical problems.
The schedule and classrooms can be consulted on the website of the degree course or in the Virtual Campus (CV) of the university. In it you will find some of the material and all the information related to this subject.
Theoretical lessons. The teacher will be developing the different parts of the program. The CV will also have available to the students a bibliography and support material, if necessary, for the theory and/or problems.
Solving problem lessons. The lists of problems to be solved will be available in the CV.
During the seminars, some concepts will be developed by the students.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|
Final exam | 45% | 3 | 0.12 | 2, 5, 1, 9, 8, 7, 6, 3, 4 |
Partial exam | 35% | 3 | 0.12 | 5, 9, 3, 4 |
Second-chance Examination | 45% | 0 | 0 | 9 |
Seminaris (3 activities) | 20% | 6 | 0.24 | 2, 1, 9, 8, 7, 3 |
See the Catalan version.
See the Catalan version.
Name | Group | Language | Semester | Turn |
---|---|---|---|---|
(PAUL) Classroom practices | 1 | Catalan | first semester | morning-mixed |
(SEM) Seminars | 1 | Catalan | first semester | morning-mixed |
(TE) Theory | 1 | Catalan | first semester | morning-mixed |