Dynamical Systems

The field of Dynamical Systems originated in the difficult mathematical questions related to the movements of the planets and the moon: Are there periodic orbits? Will the solar system keep its present form in the distant future or could it happen that one of the planets leave the system? Or, could it come to a collision between planets?

The mathematical theory of dynamical systems provides concept, ideas and tools, in ordfer to analyse and model dynamical processes in all fields of natural sciences, making use of nearly all branches of mathematics.

The aim of this class is to give a (biased) introduction to dynamical systems, presenting a few topics that are directly related to current field of research of members of the Department.

We shall start by discussing simple models of discrete dynamical systems in which the dynamics is determined by the iteration of a map. There will be example of "minimal", "transitive", "structurally stable" and "ergodic" systems. We shall then move to the study of invariant manifolds of hyperbolic points and hyperbolic sets. The next topic is gradientlike flows which will allow to introduce Morse theory. Finally, time permitting, we shall look at Hamiltonian vector fields.

The goal of this class is to be prolegomenon to the theory of Dynamical Systems and present interesting ideas and techniques which can be transposed to other fields.

Connaissances et compétences pré-requises ou co-requises

A Bachelor in Mathematics is the only pre-requisite.

Theoretical classes
Exercices to do at home

Contacts

jean.gutt@ulb.be

Campus

Plaine

Méthode(s) d'évaluation

• Examen oral
• Autre

Examen oral

• Examen hors-session

• Question ouverte à réponse courte
• Question ouverte à développement long
• Question visuelle

Autre

Langue(s) d'évaluation

• anglais
• français
• (éventuellement Allemand )