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.).

Goals

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.

Assessment method

Individual project with an oral presentation and a final report. 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. Quality of peer reviewing could also be considered in the evaluation.

Language of instruction

Anglais