Sistemas Dinâmicos

Objectives

The goal of this course is to introduce the language and the techniques of the theory of finite dimensional dynamical systems. We study the qualitative theory of ordinary differential equations and of difference equations (iteration of maps). We analyse the geometric and computational aspects of dynamical systems, including the concept of chaos. We analyse several applications of the dynamical systems theory.

Syllabus

I. Introduction to dynamical systems. 1.1 Finite dimensional dynamical systems. 1.2 Differential equations as dynamical systems. 1.3 Difference equations as dynamical systems. 1.4 Classification of fixed points. 1.5 Hamiltonian dynamical systems. 1.6 Numerical methods. 1.7 Maps of an interval and strange attractors. 1.8 The stable, center and unstable manifolds. 1.9 Cellular automata. 1.10 Maps of the complex plane. Fractal sets. II. Qualitative theory of dynamical systems. 2.1 Limit sets. 2.2 Poincaré maps. 2.3 The Poincaré-Bendixon theorem. 2.4 Dynamics in the centre manifold. 2.5 Bifurcations. III. Chaos. 3.1 The Smale horseshoe. 3.2 Cantor sets. 3.3 Chaos and sensitivity to initial conditions. 3.4 Lyapunov exponents and the Oseledets theorem. IV. Applications of the theory of dynamical systems 4.1 Celestial mechanics. 4.2 Biological systems. 4.3 Control theory and applications. 4.4 Synchronization.

Prerequisites

To reach the goals of the course, the students must have previous knowledge of basic calculus, ordinary differential equations and computer programming.

Cross Competence Component

Students are expected to develop their oral communication and critical thinking skills. This component corresponds to 20% of the evaluation.

Laboratorial Component

Belousov-Zhabotinski experiment, 0.05 ECTS.

Programming And Computing Component

Problems asessment has a strong computing component that accounts for 90% of th work of th problem series.

Ethical Principles

All members of a group are responsible for the group’s work In any assessment, every student shall honestly disclose any help received and sources used. In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.