Analytic classification of saddle nodes

Voronin, S.; Meshcheryakova, Yu.

doi:10.1090/S0077-1554-05-00151-2

Analytic classification of saddle nodes
HTML articles powered by AMS MathViewer

by S. M. Voronin and Yu. I. Meshcheryakova
Translated by: G. G. Gould

Trans. Moscow Math. Soc. 2005, 85-103

DOI: https://doi.org/10.1090/S0077-1554-05-00151-2

Published electronically: November 9, 2005

PDF | Request permission

Abstract:

Isolated degenerate elementary singular points (saddle nodes) of germs of holomorphic vector fields in $(\mathbb {C}^2,0)$ are studied. An analytic classification of them is obtained; it is shown that the analytic classification has two times more moduli (numeric and functional) than the orbital analytic classification. A theorem on sectorial normalization is proved.

References

Yu. S. Il′yashenko, Nonlinear Stokes phenomena, Nonlinear Stokes phenomena, Adv. Soviet Math., vol. 14, Amer. Math. Soc., Providence, RI, 1993, pp. 1–55. MR 1206041, DOI 10.1016/j.cnsns.2015.04.003
Jean Martinet and Jean-Pierre Ramis, Problèmes de modules pour des équations différentielles non linéaires du premier ordre, Inst. Hautes Études Sci. Publ. Math. 55 (1982), 63–164 (French). MR 672182
Jean Martinet and Jean-Pierre Ramis, Classification analytique des équations différentielles non linéaires résonnantes du premier ordre, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 4, 571–621 (1984) (French). MR 740592
Loïc Teyssier, Équation homologique et cycles asymptotiques d’une singularité nœud-col, Bull. Sci. Math. 128 (2004), no. 3, 167–187 (French, with English and French summaries). MR 2045100, DOI 10.1016/j.bulsci.2003.10.002
L. Teyssier, Analytical classification of singular saddle-node vector fields, J. Dynam. Control Systems 10 (2004), no. 4, 577–605. MR 2095942, DOI 10.1023/B:JODS.0000045365.56394.b4
S. M. Voronin, The Darboux-Whitney theorem and related questions, Nonlinear Stokes phenomena, Adv. Soviet Math., vol. 14, Amer. Math. Soc., Providence, RI, 1993, pp. 139–233. MR 1206044
S. M. Voronin and A. A. Grintchy, An analytic classification of saddle resonant singular points of holomorphic vector fields in the complex plane, J. Dynam. Control Systems 2 (1996), no. 1, 21–53. MR 1377427, DOI 10.1007/BF02259621
A. A. Grintchy, Analytic classification of saddle resonant singular points of holomorphic vector fields in the complex plane, Dep. in VINITI 24.05.96. no. 1690 B96, 24 C . (Russian)
S. M. Voronin and Yu. I. Meshcheryakova, Analytic classification of generic degenerate elementary singular points of germs of holomorphic vector fields on the complex plane, Izv. Vyssh. Uchebn. Zaved. Mat. 1 (2002), 13–16 (Russian); English transl., Russian Math. (Iz. VUZ) 46 (2002), no. 1, 11–14. MR 1896423
Yu. I. Meshcheryakova, Formal classification of degenerate elementary singular points, Sobolev-type equations (Russian), Chelyab. Gos. Univ., Chelyabinsk, 2002, pp. 197–206 (Russian). MR 1893717