| 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 |
(i) To understand the rudiments of one-dimensional and two-dimensional dynamical systems theory, both in continuous and discrete-time context;
(ii) To use the tools of the theory of one-dimensional and two-dimensional dynamic systems to analyze a given dynamical system;
(iii) To recognize some famous examples of one-dimensional and two-dimensional dynamical systems;
(iv) To analyze and understand mathematical proofs, particularly in the context of dynamical systems theory;
(v) To apply Dynamical Systems theory to several mathematical models;
(vi) To communicate using mathematical language, written and orally.
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Syllabus |
1. Unidimensional discrete dynamics
1.1 Attractive and repulsive periodic points
1.2 Quadratic family and one-sided symbolic dynamics
1.3 Topological conjugation and structural stability
1.4 Circle homeomorphisms
1.5 Morse-Smale diffeomorphisms
1.6 Von Neumann ergodic Theorem
2. Continuous bidimensional dynamics
2.1 Linear flows. Flows in S2 and T2
2.3 Lyapunov stability
2.4 Poincaré-Bendixson Theorem
2.5 Lotka-Volterra systems
2.6 Gradient vector fields
2.7 Hyperbolicity and Andronov-Pontryagin stability
2.8 Hartman-Grobman theorem (statement)
2.9 Stable manifold theorem (statement)
2.10 Poincaré recurrence theorem
2.11 Birkhoff’s ergodic theorem
3. Discrete bidimensional dynamics
3.1 Hyperbolicity and Andronov-Pontryagin stability
3.2 Hartman-Grobman theorem (statement)
3.3 Stable manifold theorem (statement)
3.4 Bilateral symbolic dynamics
3.5 Smale horseshoe
3.6 Anosov automorphisms
3.7 Solenoid attractor
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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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