Course 2024-2025

Advanced methods for non-linear systems [SMATM227]

  • 6 credits
  • 30h+30h
  • 2nd quarter
Language of instruction: French / Français

Learning outcomes

By the end of the course, students will be familiar with original techniques for studying dynamical systems and should be able to use these techniques in real-life applications (e.g. neuroscience, power grids, finance, etc.).

Objectives

Studying nonlinear (dynamical) systems is crucial in many scientific disciplines, but only few techniques provide a global and systematic approach to those systems. This course will introduce such techniques, which are based on advanced mathematical tools (e.g. operator theory). Emphasis will also be put on the use and development of numerical methods.

Content

The course will mainly focus on operator-theoretic methods, which allow to turn a nonlinear system into a linear (but infinite-dimensional) system. However the exact content of the course might change from one year to another, depending on the students’ interest as well as recent developments in research.

Table of contents

Dynamical systems theory : general reminder (attractor, stability, chaos), ergodic theory.

Operator theory : Koopman and Perron-Frobenius operators, spectral decomposition, interplay between geometric invariants and spectral properties.

Numerical methods : computation of spectrum and eigenfunctions (Fourier averages, DMD algorithm, Arnoldi method), projection of an operator on a basis.


Teaching methods

First part: lectures on theory and numerical methods (in english)

Second part: discussion(s) and work on the project (see below). Active participation of students is encouraged. Tutorials will also be organised.

Evaluations

Individual project with an oral presentation (in French) and a final report (in English). In his/her project, the student will use the techniques presented in the course, focusing on a specific application chosen according to his/her preferences. The project will be evaluated by the course holder and/or the subsitute teacher. Quality of peer reviewing could also be considered in the evaluation.

Language of instruction

French / Français

Location for course

NAMUR

Organizer

Faculté des sciences
Rue de Bruxelles, 61
5000 NAMUR

Degree of Reference

Master's Degree