# zbMATH — the first resource for mathematics

The analytic and formal normal form for the nilpotent singularity. The case of generalized saddle-node. (English) Zbl 1027.34099
The generalized saddle-node case of certain vector fields is considered. The aim is to provide analogous results to those of the same author and H. Żolądek [J. Differ. Equations 179, 479–537 (2002; Zbl 1005.34034)], and of other authors cited in the current paper, in the generalized cusp case. It is shown that two germs of analytic vector fields (with generalized saddle-node singularity) are analytically equivalent if and only if their monodromy groups are analytically equivalent. The proof makes use of a restricted version of the same theorem due to M. Berthier, R. Meziani and P. Sad [Bull. Sci. Math. 123, 351–370 (1999; Zbl 0963.32021)].

##### MSC:
 34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
Full Text:
##### References:
 [1] Arnold, V.I.; Varchenko, A.N.; Gusein-Zade, S.M., Singularities of differentiable mappings, Monographs in mathematics, 82, (1985), Birkhäuser Boston, Russian: v. 1 Nauka, Moscow, 1982 [2] Berthier, M.; Meziani, R.; Sad, P., On the classification of nilpotent singularities, Bull. sci. math., 123, 5, 351-370, (1999) · Zbl 0963.32021 [3] Cerveau, D.; Moussu, R., Groupes d’automorphismes de $$(C,0)$$ et équations différentielles $$ydy+⋯=0$$, Bull. soc. math. France, 116, 459-488, (1988) · Zbl 0696.58011 [4] Elizarov, P.M.; Il’yashenko, S.Yu.; Shcherbakov, A.A.; Voronin, S.M., Finitely generated groups of germs of one-dimensional conformal mappings and invariants for complex singular points of analytic foliations of complex plane, (), 57-105 · Zbl 1010.32501 [5] Il’yashenko, Yu.S., Nonlinear Stokes phenomena, (), 1-55 · Zbl 0804.32011 [6] Loray, F., Réduction formelle des singularités cuspidales de champs de vecteurs analytiques, J. differential equations, 158, 152-173, (1999) · Zbl 0985.37014 [7] Loray, F.; Meziani, R., Classification de certaines feuilletages associés a un cusp, Bol. soc. bras. mat., 25, 93-106, (1984) [8] Martinet, J.; Ramis, J.-P., Probleme de modules pour des équations différentielles non lineaires du premier ordre, Publ. math. IHES, 55, 63-164, (1982) · Zbl 0546.58038 [9] Martinet, J.; Ramis, J.-P., Classification analytique des équations différentielles resonantes du premier ordre, Ann. sci. école, norm. sup., 16, 571-621, (1983) [10] Meziani, R., Classification analytique d’équations différentielles $$ydy+⋯=0$$ et espace de modules, Bol. soc. bras. mat., 21, 23-53, (1996) · Zbl 0880.34002 [11] R. Meziani, Classification analytique d’équations différentielles $$ydy+⋯=0$$ et espace de modules, Thése, l’Université Ibn Tofaile, 1998 [12] Ramis, J.-P., Confluence et résurgence, J. fac. sci. univ. Tokyo sec. IA. math., 36, 706-716, (1989) [13] E. Stróżyna, H. Żoła̧dek, The analytic and formal normal forms for the nilpotent singularity, J. Differential Equations (in press) · Zbl 1005.34034 [14] Takens, F., Singularities of vector fields, Publ. math. IHES, 43, 47-100, (1974) · Zbl 0279.58009 [15] Xianghong, G., Integrable analytic vector field with a nilpotent linear part, Ann. inst. Fourier, 45, 1449-1470, (1995) · Zbl 0835.58032
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.