×

Reversibility and classification of nilpotent centers. (Réversibilité et classification des centres nilpotents.) (French) Zbl 0803.34005

We consider a germ \(\omega\) of analytic 1-form in \(\mathbb{R}^ 2\), 0 with 1- jet \(ydy\). We prove that if \(\omega= 0\) defines a center (i.e. all solutions are cycles) there exists an analytic involution of \(\mathbb{R}^ 2\), 0 preserving the phase portrait of the system. Geometrically this means that analytic nilpotent centers are built by pull back with fold applications. A theorem of equivariant conjugacy leads to the complete classification of such centers.
Reviewer: M.Berthier (Dijon)

MSC:

34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34A26 Geometric methods in ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34A99 General theory for ordinary differential equations
32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)
32S65 Singularities of holomorphic vector fields and foliations
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] M. BRUNELLA, Vanishing holonomy and monodromy of certain centers and foci, preprint SISSA (1992). · Zbl 0841.32020
[2] D. CERVEAU et R. MOUSSU, Groupes d’automorphismes de (ℂ,0) et équations différentielles y dy+ = 0, Bull. Soc. Math. France, 116 (1988), 459-488. · Zbl 0696.58011
[3] H. DULAC, Sur LES cycles limites, Bull. Soc. Math. France, 51 (1923), 45-188. · JFM 49.0304.01
[4] H. DULAC, Recherche sur LES points singuliers des équations différentielles, J. École Polytechnique, 2, sec. 9 (1904), 1-125. · JFM 35.0331.02
[5] J. ÉCALLE, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Actualités Mathématiques, Publication Hermann, 1992. · Zbl 1241.34003
[6] Y. IL YASHENKO, Finiteness theorem for limit cycles, Translations of Mathematical Monographs, vol. 94, AMS, 1991. · Zbl 0743.34036
[7] A.M. LIAPOUNOV, Stability of motion, Academic Press, 1966. · Zbl 0161.06303
[8] J. MARTINET et J.-P. RAMIS, Analytic classification of resonant saddles and foci, Singularities and Dynamical Systems, North-Holland Math. Studies, 103 (1985), 109-135. · Zbl 0596.34021
[9] J.F. MATTÉI et R. MOUSSU, Holonomie et intégrales premières, Ann. Sci. École Normale Supérieure, 13 (1980), 469-523. · Zbl 0458.32005
[10] R. MÉZIANI, Problème de modules pour des équations dégénérées de ℂ2, Thèse de l’Université de Rennes, 1992.
[11] R. MOUSSU, Une démonstration géométrique d’un théorème de Poincaré-liapounov, Astérisque, 98-99 (1982), 216-223. · Zbl 0523.34039
[12] R. MOUSSU, Symétries et formes normales des centres et foyers dégénérées, Ergod. Th. and Dynam. Systems, 2 (1982), 241-251. · Zbl 0509.34027
[13] R. MOUSSU, Holonomie évanescente des équations différentielles dégénérées transverses, in Singularities and dynamical systems, North Holland, 1985. · Zbl 0569.58012
[14] H. POINCARÉ, Mémoires sur LES courbes définies par une équation différentielle, J. Math. Pures Appl., 37 (1881), 375-442. · JFM 13.0591.01
[15] G. SANSONE et R. CONTI, Non linear differential equations, Pergamon Press, 1964. · Zbl 0128.08403
[16] F. TAKENS, Singularities of vector fields, Publ. Math. IHES, 43 (1974), 47-100. · Zbl 0279.58009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.