Intitulé français du sujet de thèse proposé : Forme normale d'équations différentielles contraintes
Intitulé anglais du sujet proposé : Normal form of differential equations with constraints
Unité de recherche : UMR 5219
Domaine scientifique principal de la thèse : Mathématiques
Domaine scientifique secondaire de la thèse : Mathématiques et interactions
Thème et sous-thème prioritaire : 1.a (Nouvelles synergies en mathématiques); 1.e-1.h (systèmes dynamiques et contrôle)
Nom, prénom et courriel du directeur de thèse : STOLOVITCH, Laurent, stolo@picard.ups-tlse.fr
2/ Projet de thèse en anglais
The goal of this project is to study systems of differential equations which are unsolved with respect to the derivative : F(x,y,y')=0 where y is the unknown map of the variable x with values in a p-dimensional complex vector space and where y' denotes the derivative of y with respect to x. The map F takes its values in an s-dimensional complex vector space. The main goal of this research program is to find local models (called normal forms) of these equations in a neighbourhood of a point which can be obtained by the mean of a change of variables of the form (X,Y)=G(x,y). We then expect to be able to study the dynamical properties of the normal form more easily and to pull-back them back by the mean of the transformation.
Particular cases of this problem are the problem of singularity of differential equations y'=f(y) as well as the problem of differential equations with constraints: y'=f(x,y) et g(x,y)=0.
This project is ambitious and stands at the crossroad of the theory of dynamical systems as developed since Poincaré by Arnold, Moser, ... and of the theory of singularities as developed by Thom, Arnold,... and of the problems of control theory as well..