Complete transversal and formal normal forms of germs of vector fields
DOI:
https://doi.org/10.5269/bspm.46354Abstract
In this work, inspired by the technique of the complete transversal, used for the classification of plane branches, developed by Hefez, A. and Hernandes, M., as well as Bruce, J.W., Kirk, N.P. and du Plesis, A.A., study the singularities of applications, we establish a classification of vector fields through their normal forms. In the case of vector fields with non zero linear part in $(\mathbb{C}^{2}, 0) $ and nilpotent fields in $(\mathbb {C}^{n}, 0), n\geq 2$ we recover the classical normal forms for those fields, and we provide a formal normal form different from Takens in dimension 2. Likewise, we obtain the normal form for the vector fields in $(\mathbb{C},0)$ of any multiplicity.
References
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2. Dulac, H., Recherches sur les points singuliers des equations differentielles. J. Ecole Polytechnique 2, 1-125, (1904). https://doi.org/10.5802/afst.267
3. Hefez, A. and Hernandes, M., The analytic classification of plane branches. Bull. London Math. Soc. 43, 289-298, (2011). https://doi.org/10.1112/blms/bdq113
4. Ichikawa, F., Finitely determined singularities of formal vector fields. Invent. Math. 66, 199-214, (1982). https://doi.org/10.1007/BF01389391
5. Ichikawa, F., Classification of finitely determined singularities of formal vector fields on the plane. J. Math. 8 N.2, 463-472, (1985). https://doi.org/10.3836/tjm/1270151227
6. Loray, F., Feuilletages Holomorphes a' Holonomie R'esoluble. Th'ese pour obtenir le grade de Docteur de l, Universite de Rennes I, (1994).
7. Loray, F., Reduction formelle des singularit'es cuspidales de champs de vecteurs analytiques. Journal of Differential Equations. 58, 152-173, (1999). https://doi.org/10.1016/S0022-0396(99)80021-7
8. Manoel, M. and Zeli, I., Complete transversals of symmetric vector fields. Journal of Singularities. 12, 124-130, (2015). https://doi.org/10.5427/jsing.2015.12h
9. Paul, E., Feuilletages holomorphes singuliers a' holonomie r'esoluble. J. Reine Angew. Math. 514, 9-70, (1999). https://doi.org/10.1515/crll.1999.074
10. Paul, E., Formal normal forms for the perturbations of a quasi-homogeneous Hamiltonian vector field. Journal of Dynamical and Control Systems. 10 N.4, 545-575, (2004). https://doi.org/10.1023/B:JODS.0000045364.52822.a0
11. Poincare, H., Premiere These-Sur les Propietes des Fonctions d'efinies par les Equations aux Differences Partielles. 'These presentees a la Faculte des Sciences de Paris pour obtener le Grade de Docteures Sciences Math'ematiques. Gauthier-Villars, (1879).
12. Ramırez-Carrasco S., Transversal Completa y Formas Normales Formales de Campos Vectoriales. Doctoral Thesis presented to the Pontificia Universidad Catolica del Peru, (2015).
13. Stroyna, E. and Zoladek., H., The analytic and formal normal forms for the nilpotent singularity. J. Differential Equations 179, 479-537, (2002). https://doi.org/10.1006/jdeq.2001.4043
14. Strozyna, E., Orbital formal normal forms for general Bogdanov-Takens singularity. J. Differential Equations 193 N.1, 239-259, (2003). https://doi.org/10.1016/S0022-0396(03)00137-2
15. Strozyna. E. and Zoladek, H., Multidimensional formal Takens normal form. Bull. Belg. Math. Soc. Simon Stevin 15, 927- 934, (2008). https://doi.org/10.36045/bbms/1228486416
16. Strozyna, E. and Zoladek, H., Divergence of the reduction to the multidimensional nilpotent Takens normal form. Nonlinearity 24, 3129-3141, (2011). https://doi.org/10.1088/0951-7715/24/11/007
17. Strozyna, E. and Zoladek, H., The complete formal normal form for the Bogdanov-Takens singularity. Moscow Mathematical Journal 15, Issue 1, 141-178, (2015). https://doi.org/10.17323/1609-4514-2015-15-1-141-178
18. Takens, F., Singularities of vector fields. Inst. Hautes Etudes Sci. Publ. Math. 43, 47-70, (1974). https://doi.org/10.1007/BF02684366
19. Zariski, O., Le Probleme des Modules pour les Branches Planes. Cours donn'e au Centre de Math'ematiques de L'Ecole Polytechnique. Nouvelle ed. revue par l'auteur. Redige par Francois Kimety et Michel Merle. Avec un appendice de Bernard Teissier. Paris, Hermann, (1986). English translation by Ben Lichtin: The Moduli Problem for Plane Branches. University Lecture Series, Vol. 39, AMS (2006). https://doi.org/10.1090/ulect/039
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2022-12-26
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Pontificia Universidad Católica del Perú
Grant numbers DGI-2016-1-0069
