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

Berthier, M.; Moussu, R.

Ann. Inst. Fourier 44, No. 2, 465-494 (1994).

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

Keywords:

center; analytic involution; phase portrait; equivariant conjugacy; classification

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