| Code |
14805
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| Year |
3
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| Semester |
S2
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| ECTS Credits |
6
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| Workload |
TP(60H)
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| Scientific area |
Mathematics
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Entry requirements |
NA
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Learning outcomes |
The objectives of the course are:
(i) to understand the basic concepts of one-dimensional and two-dimensional dynamical systems in both discrete and continuous time;
(ii) to use tools from this theory to analyse the qualitative behaviour of dynamical systems;
(iii) to recognise and study some classical examples of one-dimensional and two-dimensional dynamical systems;
(iv) to analyse and understand mathematical proofs in the context of dynamical systems theory; (v) to apply concepts and results from this theory in the modelling of phenomena described by dynamical systems;
(vi) to communicate, both orally and in writing, using appropriate mathematical language.
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Syllabus |
1. Fundamentals and Examples
1.1 The ingredients of dynamics
1.2 Discrete-time dynamical systems
1.3 Examples of discrete-time dynamical systems
2. One-Dimensional Discrete Dynamics
2.1 Hyperbolicity
2.2 The quadratic family
2.3 Topological conjugacy
2.4 Chaos
2.5 Structural stability
2.6 Periodic points and Sharkovsky’s Theorem
2.7 Bifurcations
2.8 Circle homeomorphisms
2.9 Morse–Smale diffeomorphisms
3. Multidimensional Discrete Dynamics
3.1 Linear multidimensional dynamics
3.2 Structural stability
3.3 Hartman–Grobman Theorem
4. Two-Dimensional Continuous-Time Dynamics
4.1 Hyperbolic linear dynamical systems
4.2 Hartman–Grobman Theorem
4.3 Lyapunov stability
4.4 Poincaré–Bendixson Theorem
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Main Bibliography |
- Colonius, F., & Kliemann, W. (2014). Dynamical Systems and Linear Algebra. Graduate Studies in Mathematics, 158. American Mathematical Society.
- Doering, C. I., & Lopes, A. O. (2016). Equações Diferenciais Ordinárias. Coleção Matemática Universitária. (6.ª edição). IMPA.
- Hirsch, M. W., Smale, S., & Devaney, R. L. (2013). Differential Equations, Dynamical Systems, and an Introduction to Chaos. (3.ª edição). Elsevier.
- Katok, A., & Hasselblatt, B. (2005). A Moderna Teoria de Sistemas Dinâmicos. Lisboa: Fundação Calouste Gulbenkian.
- Robinson, C. (1999). Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. Studies in Advanced Mathematics. (2nd edition). CRC Press.
- Sternberg, S. (2010). Dynamical Systems. Dover Books on Mathematics.
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Teaching Methodologies and Assessment Criteria |
Knowledge assessment during the Teaching-Learning process will consist of five individual sets of exercises and problems to be completed during the semester and two written tests.
Grades obtained in the two written tests (T1 and T2) are presented on a scale of 0 to 20 points, and each set of exercises (Ei) on a scale of 0 to 1 point. The final Teaching-Learning grade will be determined by rounding the result obtained from the following calculation:
CF = 0.75T + E
where T = (T_1 + T_2)/2 and E = E1 + E2 + E3 + E4 + E5, provided it is less than or equal to 16 points. If after rounding the grade exceeds 16 points, the student must undergo an oral examination, and the final grade, in this case, will be assigned by the jury of the respective examination, ensuring it is not lower than 16 points nor higher than the CF grade.
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Language |
Portuguese. Tutorial support is available in English.
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