zbMATH — the first resource for mathematics

Integrable analytic vector fields with a nilpotent linear part. (English) Zbl 0835.58032
Summary: We study the normalization of analytic vector fields with a nilpotent linear part. We prove that such an analytic vector field can be transformed into a certain form by convergent transformations when it has a non-singular formal integral. We then prove that there are smoothly linearizable parabolic analytic transformations which cannot be embedded into the flows of any analytic vector fields with a nilpotent linear part.

37G05 Normal forms for dynamical systems
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
32S65 Singularities of holomorphic vector fields and foliations
Full Text: DOI Numdam EuDML arXiv
[1] V.I. ARNOL’D and Yu. S. IL’YASHENKO, Ordinary differential equations, in “Dynamical Systems I, EMS” vol. 1, Springer-Verlag, Berlin, 1990. · Zbl 0789.53017
[2] A. BAIDER and J. C. SANDERS, Further reduction of the Takens-bogdanov normal form, J. Diff. Equations, 99 (1992), 205-244. · Zbl 0761.34027
[3] R.I. BOGDANOV, Versal deformation of a singularity of a vector field on the plane in the case of zero eigenvalues, Seminar Petrovski (1976), and Selecta Math. Soviet, n° 4, 1 (1981), 389-421. · Zbl 0518.58030
[4] D. CERVEAU and 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
[5] X. GONG, Divergence for the normalization of real analytic glancing hypersurfaces, Commun. Partial Diff. Equations, 19, n° 3 & 4 (1994), 643-654. · Zbl 0804.53080
[6] R.B. MELROSE, Equivalence of glancing hypersurfaces, Invent. Math., 37 (1976), 165-191. · Zbl 0354.53033
[7] F. TAKENS, Singularities of vector fields, Publ. Math. I.H.E.S., 43 (1974), 47-100. · Zbl 0279.58009
[8] S.M. WEBSTER, Holomorphic symplectic normalization of a real function, Ann. Scuola Norm. Sup. di Pisa, 19 (1992), 69-86. · Zbl 0763.58010
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.