Nonlinear dynamic systems

Disciplinary field: Methods (Mandatory)
Level: M2
Credits: 3 ECTS

Teachers: Guillaume Lapeyre (LMD) et Davide Faranda (LSCE)
Teaching type: Cours/TD/TP
Hourly volume: 30h

Evaluation: L’évaluation se fera par un devoir maison (25% de la note finale), un compte-rendu de TP (5%) et une étude d’un système dynamique connu en géophysique  (70%).

Keywords: Systèmes nonlinéaires, chaos, prévisibilité, bifurcations
Prerequisites: Base de l’analyse mathématique: équations aux dérivées ordinaires et partielles. Bases de l’analyse mathématique : équations aux dérivées partielles. Bases de l’algèbre linéaire : calcul matriciel, espace vectoriel. Bases de la théorie des probabilités. Programmation matlab ou python.

A large part of geophysical phenomena are generally described by systems of non-linear equations. Even if the laws of evolution are known, they do not necessarily imply a purely deterministic evolution of the system. This is one of the reasons for the difficulty of weather or climate predictions.

Henri Poincaré in 1890 was one of the first to highlight this non-deterministic behavior by studying the equations of motion of the planets of the solar system. In 1963, Ed Lorenz showed the chaotic nature of the atmosphere and the existence of an associated « strange attractor ».

The objective of this course is to introduce the basic mathematical concepts allowing characterizing qualitatively and quantitatively non-linear dynamical systems: in terms of their stability, predictability and chaotic nature, but also in terms of statistics of extreme events associated with them. Exemples of the relevence of dynamical systems metrics for the description and the quantitative analysis of atmospheric motions will be given with a focus on atmospheric dynamics. Practical work will allow the students to manipulate and acquire these notions using simple models.