Dynamical Systems

Code	14805
Year	3
Semester	S2
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	NA
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.
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
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.
Language	Portuguese. Tutorial support is available in English.