Equações Diferenciais Ordinárias

Objectives

To provide a high level introduction to the Theory of Ordinary Differential Equations, with emphasis on the study of geometric and topological properties, and namely of fundamental concepts and results of qualitative theory, hyperbolicity, and stability.

Syllabus

Basic notions: existence, uniqueness, regularity and extension of solutions; continuous dependence with respect to the initial conditions. Geometric theory: phase portraits; homoclinic, heteroclinic and periodic orbits; invariant sets;limit sets; Poincaré-Bendixson's theorem. Linear equations: phase portraits; linear variational equation; periodic coefficients; attracting, repelling and hyperbolic systems; linear, topological and differentiable conjugation. Hyperbolicity: hyperbolic fixed points; Grobman- Hartman theorem: topological conjugacy for diffeomorphisms and vector fields. Stability: stability and asymptotic stability; Lyapunov functions; stability and instability criteria; mechanical systems. Index theory: index theory for vector fields in the plane; Brouwer's fixed point theorem; index of isolated critical points. Bifurcation theory: bifurcation diagrams; homological equations and normal forms; central manifold theorem; stability of critical points.

Prerequisites

Differential and Integral Calculus III.

Cross Competence Component

The UC allows the development of transversal competences on Critical Thinking, Creativity and Problem Solving Strategies, in class, in autonomous work and in the several evaluation components. The percentage of the final grade associated with these competences should be around 15%.

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.