Dynamical Systems

Code	14805
Year	3
Semester	S2
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	NA
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.
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
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.
Teaching Methodologies and Assessment Criteria	Continuous assessment is based on two written tests and five individual lists of exercises and problems to be completed throughout the semester. The two tests (T1 and T2) are graded on a scale from 0 to 20, while each exercise list (Ei) is graded on a scale from 0 to 1. The final continuous assessment grade is calculated as: CF = 0.75T + E, where T = (T1 + T2)/2 and E = E1 + E2 + E3 + E4 + E5. The final grade is obtained by rounding the result to the nearest whole number. If the rounded grade is higher than 16, the student must take an oral exam. In this case, the final grade is assigned by the examination panel and cannot be lower than 16. All students are admitted to the final exam. Students with special status are subject to specific rules defined by the Academic Regulations.
Language	Portuguese. Tutorial support is available in English.