Dynamical Systems
Code: 100118Credits: 6
| Degree programme | Type | Course |
|---|---|---|
| Mathematics | OP | 4 |
Contact lecturer
- Name :
- Jordi Villadelprat Yague
- Email :
- jordi.villadelprat@uab.cat
Group languages
You can consult this information at the end of the document.
Prerequisites
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.
Objectives
This course is an introduction to the modern theory of dynamical systems. The first objective is for students to become familiar with the abstract notion of a dynamical system and the basic concepts of this theory: stability, attractor, invariant sets, omega limits, etc. The second objective is to understand the local behavior of both discrete and continuous dynamical systems in the vicinity of an equilibrium point or a periodic orbit. This local behavior is based on the topological classification of linear systems in ℝⁿ, both those derived from the flow of ordinary differential equations (continuous dynamical systems) and those arising from function iteration (discrete dynamical systems). Linear systems are very important because, on the one hand, they appear in the study of many relevant physical phenomena, and on the other hand, they represent the first approximation to more complex systems.
The qualitative theory of differential equations began with the work of Poincaré around 1880, in the context of his studies in celestial mechanics. It aims to understand properties of solutions without the need to solve the equations, among other reasons because exact solutions are only possible in exceptional cases. This qualitative approach, when combined with appropriate numerical methods, can in some cases be equivalent to having the solutions of the equation. The course will deepen the knowledge and study—introduced in previous subjects—of the qualitative theory of differential equations in higher-dimensional spaces, with an emphasis on the local structure of equilibrium points (both degenerate and non-degenerate) and the stability of periodic orbits.
The final objective of the course is to introduce techniques for understanding global discrete dynamics. The guiding thread will be a parametric family of discrete dynamical systems: unimodal maps, which (for certain parameter values) exhibit dynamics that naturally lead to the notion of chaos. For these systems, numerical approximation is not feasible, and new tools are required to understand their dynamics. Chaotic systems often appear in applications (e.g., weather prediction problems, electrical circuits, etc.).
Learning outcomes
- Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
- Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
- Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
- Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
- Actively demonstrate high concern for quality when defending or presenting the conclusions of one's work.
- Apply critical spirit and thoroughness to validate or reject both one's own arguments and those of others.
- Know how to solve certain theoretical problems and be understand the existence of certain open problems in the theory of partial derivative equations and dynamical systems theory.
- Know how to apply the dynamical tools described in theory lectures to describe processes governed by differential equations.
- Know how to demonstrate the results of partial derivative equations and dynamical systems.
Contents
1. Dynamical systems on topological spaces.
- Dynamical systems defined by differential equations and by difeomorphisms.
- Orbits; Critical points and periodic orbits.
- Invariant sets and limit sets.
- Attractors. Liapunov stability.
- Conjugation of dynamic systems.
2. Study of local dynamics, discrete and continuous in R^n.
- Phase portraits near equilibrim and regular points.
- Topological classification of continuous and discrete linear systems.
- Stability (Liapunov's functions)
- Hartman theorem, stable manifold theorem and center manifold theorem.
- Periodic orbits: Poincaré map and stability.
3. Global dynamics in continuous systems.
- Ordinary differential equations in R^2 (Poincaré-Bendixson Theorem, Bendixson-Dulac theorem, existence and uniqueness of limit cycles, ...)
- Ordinary differential equations in dimension greater than 2.
4. Global dynamics in discrete systems.
- Iteration in dimensions 1 and 2.
- Unimodal applications.
- Chaos. Bernoulli's shift. Smale's Horseshoe.
Note: The contents will be adapted to the topics covered in the third-year differential equations subjects in order to avoid repetition.
Learning activities and methodology
| Title | Hours | ECTS | Learning outcomes |
|---|---|---|---|
| Exam Preparation | 28 | 1.12 | 1, 2, 3, 4, 5, 6, 7, 8, 9 |
| Study of the theoretical part | 27 | 1.08 | 1, 2, 3, 4, 5, 6, 8, 9 |
| Problem solving classes | 14 | 0.56 | 1, 2, 3, 4, 5, 6, 7, 8, 9 |
| Theory classes | 30 | 1.2 | 1, 2, 3, 4, 6, 7, 8, 9 |
| Seminars | 6 | 0.24 | 1, 2, 3, 4, 5, 6, 7, 8, 9 |
| Problem solving | 28 | 1.12 | 1, 2, 3, 4, 5, 6, 7, 8, 9 |
The course includes two hours of theoretical classes and one hour of problem-solving classes per week. During the semester, there will also be three seminar sessions, each lasting two hours.
Schedules and classroom assignments must be consulted through the UAB's online platforms. A dedicated space for this course will be available on the Virtual Campus (VC) to provide materials and share information related to the classes.
Theoretical classes: The teaching staff will develop the topics of the syllabus in the indicated order. A bibliography and part of the supporting materials, if necessary, for both theory and problem-solving, will also be available to students on the VC.
Problem-solving classes: Problem sets will be available on the VC. Some of these problems will be worked on in class.
During the seminars, certain concepts will be explored in greater depth and developed by the students.
Assessment
Continuous assessment activities
| Title | Weight | Hours | ECTS | Learning outcomes |
|---|---|---|---|---|
| First midterm | 40% | 3 | 0.12 | 1, 2, 3, 4, 5, 6, 7, 8, 9 |
| Seminars (3 activities) | 20% | 6 | 0.24 | 1, 3, 4, 5, 6, 8 |
| Second midterm | 40% | 3 | 0.12 | 1, 2, 3, 4, 5, 6, 7, 8, 9 |
| Resist exam | 80% | 5 | 0.2 | 1, 2, 3, 4, 5, 6, 7, 8, 9 |
Continuous assessment It is organized into the following blocks, each of which will be assigned a specific weight in the final grade:
- Seminars (SEM): The reports and work assigned in the three seminar sessions will be assessed.
- First partial exam (P1): Written test at the middle of the semester.
- Second partial exam (P2): Written test at the end of the semester.
If N1=0.2SEM+0.4(P1+P2) is greater than or equal to 5, then N1 is the final grade for the subject. If N1<5, the student may sit a resit exam (R) and the final grade is N2=0.2SEM+0.8R.
Single assessment On the same day as the second partial exam of the continuous assessment, students who have previously opted for single assessment will submit the work assigned in the seminar sessions (SEM) and will sit a final exam (F) covering the entire syllabus. The grade obtained will be N3=0.2SEM+0.8F. In the event that N3<5, the same resit system as for continuous assessment will apply.
A student will be considered to obtain a Not Assessable grade if the weighted proportion of completed assessment activities is less than two thirds of those scheduled for the subject.
Note: In this subject, the use of Artificial Intelligence (AI) technologies is not permitted in any of its phases. Any work that includes AI-generated content will be considered an act of academic dishonesty and may result in a partial or total penalty on the activity grade, or more severe sanctions in serious cases.
Bibliography
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, Lições de Equações Diferenciais Ordinárias, IMPA 2025 (segona edició).
Software
Course groups and languages
The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE
| Type of teaching | Group | Language | Semester | Shift |
|---|---|---|---|---|
| (TE) Theory | 1 | Catalan | first semester | morning-mixed |
| (PAUL) Classroom practices | 1 | Catalan | first semester | morning-mixed |
| (SEM) Seminars | 1 | Catalan | first semester | morning-mixed |