Normalization of holomorphic vector fields and implicit differential equations
type: Dataset
description: The aim of the classical theory of normal forms is to simplify complicated problems by using regular changes of coordinates, in order to keep the dynamical characteristics of the system. More precisely, we consider a dynamic system said to be "elementary", like a linear part of a vector field in the neighborhood of a singular point, and we focus on a perturbation of this elementary system. Normal forms are the set of all representatives of those perturbations under the action of the group of regular transformation. They are composed of terms which caracterise the dynamics of the perturbed system, and which are called "resonances". In the first part, we try to understand the local dynamic of implicit equations of the form F(x,y,y')=0, where F is a germ of holomorphic function in a neighborhood of a singular point. To this end we use the relation between implicit systems and liouvillian vector fields. The classification by contact transformations of implicit equations come from the symplectic classification of liouvillian vector fields. We use all normal forms theory for vector fields, in complex case (Bjruno, Siegel, Stolovitch), and in real case (Sternberg), adapted to liouvillian fields with symplectic transformations. We establish classification results for implicit equations according to the dynamical invariants, and existence conditions of local solutions using normal forms. In the second part, we undertake the normalization of an analytic vector field in a neighborhood of the torus. Brjuno enunciates a theorem of normalization, under conditions of control of small divisors and integrability of the normal forms ; however he doesn't give any proof of that theorem.
identifier: NNT: 2013NICE4116
issued: 2026-05-06T22:17:30.328801
language: fr
modified: 2026-05-06T22:17:30.328806
contact point: Organization
  fn: CCSD
keyword: analytic-vector-fields
keyword: champs-de-vecteurs-analytiques
keyword: champs-de-vecteurs-liouvilliens-et-hamiltoniens
keyword: equations-differentielles-implicites
keyword: formes-normales
keyword: implicit-differential-equations
keyword: infoeu-reposemanticsdoctoralthesis
keyword: liouvilian-and-hamiltonian-vector-fields
keyword: mathmath-gmmathematics-mathgeneral-mathematics-mathgm
keyword: normal-forms
keyword: petits-diviseurs
keyword: small-divisors
keyword: theses
landing page: https://theses.hal.science/tel-00944657

Normalization of holomorphic vector fields and implicit differential equations
type: Distribution
format: HTML
issued: 2026-05-06T22:17:30.380589
modified: 2026-05-06T22:17:30.308591
access URL: https://theses.hal.science/tel-00944657

test_moissonnage_selune
type: Agent