Degree | Type | Year | Semester |
---|---|---|---|
2500149 Mathematics | OT | 4 | 0 |
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. A 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 of more complicated systems.
The qualitative theory of differential equations began with the work of Poincaré towards the year 1880 in relation to his works of Celestial Mechanics. The main idea is to know properties of the solutions without needing to solve the equations. This qualitative approach, when combined with the right numerical methods, is, in some cases, equivalent to having the solutions of the equation. We will present the basic results of the qualitative theory (Liapunov theorems, Hartman theorem and theorems of the stable and central varieties) on the local structure of equilibrum points and periodic orbits. Additionaly, in R^2 Begin in the problem of detecting the existence of periodic orbits via the Poincaré-Bendixson and Bendixson-Dulac theorems.
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 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.
3. Global dynamics in continuous systems.
4. Global dynamics in discrete systems.
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.
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 |
Continuous assessment: The partial examination (35% of the total grade) and the work in charge of the seminars (20% of the total grade)
The recovery exam will only allow you to retrieve the final exam of the semester exam (45% of the total grade). The rest is considered continuous assessment, and therefore not recoverable. You must have participated in 2/3 of the activities evaluated.
IMPORTANT: A student will be considered to have submitted to the subject if 1/2 of the continuous assessment or the final exam or the second-change examination.
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 | 100% | 0 | 0 | 9 |
Seminaris (3 activities) | 20% | 6 | 0.24 | 2, 1, 9, 8, 7, 3 |
L.H. ALVES, Sistemas Dinâmicos, Mack Pesquisa, 2006.
D.K. ARROWSMITH, C.M. PLACE, An Introduction to dynamical Systems, Cambridge University Press, 1990.
D.K. ARROWSMITH, C.M. PLACE, Dynamical Systems, differential equations, maps and chaotic behaviour, Chapman & Hall Mathematics, 1992.
R.L. DEVANEY, An introduction to chaotic dynamical systems, The Benjamin/Cummings Publishing Company, Inc., 1986.
R.L. DEVANEY, Chaos, fractals and Dynamics, Computer experiments in mathematics, Addison-Wesley, 1990.
R.L. DEVANEY, A first course in chaotic dynamical systems, Theory and Experiment, Studies in Nonlinearity, 1992.
F. DUMORTIER, J.LLIBRE and J.C. ARTES, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag Berlin, 2006.
C. FERNANDEZ, F. j. VAZQUEZ, J. M. VEGAS, Ecuaciones diferenciales y en diferencias. Sistemas Dinámicso, Thomson 2003.
J. GUCKENHEIMER, P. HOLMES, Nonlinear oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, 1993.
M. HIRSCH, S. SMALE and R. DEVANEY, Differential Equations, Dynamical Systems and an Introduction to Chaos, Elsevier Academic Press, 2004.
M.C. IRWIN, Smooth Dynamical Systems, Advanced series in Nonlinear Dynamics, vol.17, World Scientific, 2001.
S. LYNCH, Dynamical Systems with Applications using MAPLE, Birkhäuser, 2000.
L. PERKO, Differential Equations and Dynamical Systems, Springer-Verlag, 1996.
C. ROBINSON, Dynamical Systems: Stability, Symbolic Dynamics and Chaos CRC Press, 1999.
J. L. ROMERO, C. GARCIA, Modelos y Sistemas Dinámicos, Univesidad de Cádiz, 1998.
J. SOTOMAYOR, Liçoes de equacoes diferenciais ordinárias, Projecto Euclides, Gráfica Editora Hamburg Ltda., 1979.