Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-30T23:46:58.873Z Has data issue: false hasContentIssue false
  • English
  • Français

Formes normales d'équations differéntielles implicites et de champs de Liouville

Published online by Cambridge University Press:  19 September 2008

M. Manouchehri
M. Manouchehri
Affiliation:
UFR de Mathématiques, CASE 7012, Université Denis Dierot, 2, Place Jussieu, 75251 Paris, Cédex 5, France
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider the partial differential equation f(x, y(x), dy(x)) = 0, where f is a smooth real function on ℝn × ℝ × (ℝn)*. Near each singularity of the characteristic foliation, a Liouville field is associated to the equation; we classify hyperbolic germs of Liouville fields up to symplectic transformations, hence we deduce normal forms for partial differential equations up to transformations which preserve the standard contact form of ℝ2n+1. For n = 1, a theorem of Davydov enables us to deduce normal forms for such equations up to transformations of the x, y plane.

Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 4 , August 1996 , pp. 779 - 789
Copyright
Copyright © Cambridge University Press 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Chaperon, M.. Quelques outils de la théorie des actions différentiables. Troisième rencontre de géométrie du Schnepfenried, vol. 1. Astérisque 107–108 (1983), 259–275.Google Scholar
[2]Chaperon, M.. Géométrie différentielle et singularités de systèmes dynamiques. Astérisque 138–139 (1986).Google Scholar
[3]Davydov, A.. Normal form of a differential equation not solvable for the derivative in a neighborhood of a singular point. Functional Anal. Appl. 19(2) (1985), 81–89.CrossRefGoogle Scholar
[4]Sternberg, S.. On the structure of local homeomorphisms of Euclidean n-spaces II. Amer. J. of Math. 80 (1958), 623–631.CrossRefGoogle Scholar
[5]Takens, F.. Singularities of vector fields. Publications mathématiques I.H.E.S 43 (1974), 47–100.CrossRefGoogle Scholar