Orbital Formal Rigidity for Germs of Holomorphic and Real Analytic Vector Fields

Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, New York (1983)CrossRefMATHGoogle Scholar

Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps, Vol. I. The Classification of Critical Points, Caustics and Wave Fronts. Birkhäuser Boston Inc, Boston (1985)MATHGoogle Scholar

Brjuno, A.D.: Analytic form of differential equations. I, II, (Russian) Trudy Moskov. Mat. Obshch. 25, 119–262 (1971); 26, 199–239 (1972). English translation: Trans. Moscow Math. Soc. 25 (1971); 26 (1972)Google Scholar

Cerveau, D., Moussu, R.: Groups d’automorphismes de \((C, 0)\) et équations différentielles \(y \, d y + \ldots = 0\). Bull. Soc. Math. Fr. 116, 459–488 (1988)CrossRefMATHGoogle Scholar

Dulac, H.: Recherches sur les points singuliers des équations différ-entielles. Journal de l’École Polytechnique 2(9), 1–125 (1904)Google Scholar

Écalle, J.: Théorie itérative, Introduction à la théorie des invariants holomorphes. J. Math. Pures et Appl. 54, 183–258 (1975)MathSciNetMATHGoogle Scholar

Écalle, J.: Les fonctions résurgentes, Tome I, II, Orsay: Publications Mathématiques d’Orsay 81, 5. Université de Paris-Sud, Département de Mathématique (1981)Google Scholar

Elizarov, P.M., Ilyashenko, Yu S.: Remarks on the orbital analytic classification of germs of vector fields. Math. USSR Sbornik 49(1), 111–124 (1984)MathSciNetCrossRefMATHGoogle Scholar

Elizarov, P.M., Ilyashenko, Yu S., 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 the complex plane. Nonlinear Stokes Phenomena. Advances in Soviet Mathematics, vol. 14, pp. 57–105. American Mathematical Society, Providence (1993)Google Scholar

Granger, J.M.: Sur un Espace de Modules de germe de Courbe Plane, Bull. Sc. Math., \(2^{e}\) série, 3–16 (1979)Google Scholar

Gunning, R.C.: Introduction to Holomorphic Functions of Several Variables, vol. II, Local Theory, Wadsworth and Brooks/Cole, Advance Books and Software, California (1990)Google Scholar

Ilyashenko, Yu S.: Dulac’s memoir “On limit cycles” and related questions of the local theory of differential equations. (Russian) Uspekhi Mat. Nauk 40(6), 41–78 (1985)MathSciNetGoogle Scholar

Ilyashenko, Yu S.: Nonlinear Stokes phenomena. Nonlinear Stokes Phenomena. Advances in Soviet Mathematics, vol. 14, pp. 1–55. American Mathematical Society, Providence, RI (1993)CrossRefGoogle Scholar

Ilyashenko, YuS, Yakovenko, SYu.: Lectures on Analytic Differential Equations. Graduate Studies in Mathematics, vol. 86. American Mathematical Society, Providence (2008)MATHGoogle Scholar

Jaurez-Rosas, J.A.: Real-formal orbital rigidity for germs of real analytic vector fields on the real plane. J. Dyn. Control Syst. 2(22) (2016).  https://doi.org/10.1007/s10883-016-9314-y

Loray, F.: Rèduction formalle des singularities cuspidales de champs de vecteurs analytiques. J. Differ. Equ. 158, 152–173 (1999)CrossRefMATHGoogle Scholar

Martinet, J., Ramis, J.-P.: Problèmes de modules pour des équations différentielles non linéaires du premier ordre, Inst. Hautes Études Sci. Publ. Math. vol. 5, pp. 63–164 (1982)Google Scholar

Martinet, J., Ramis, J.-P.: Classification analytique des équations différentielles non linéaires résonnantes du premier ordre, Ann. Sci. École Norm. Sup. (4) 16 (1983); 4, 571–621 (1984)Google Scholar

Mattei, J.-F.: Modules de feuilletages holomorphes singuliers I. Équisingularité. Invent. Math. (French) 23(2), 297–325 (1991)CrossRefMATHGoogle Scholar

Ortiz-Bobadilla, L., Rosales-González, E., Voronin, S.M.: Rigidity theorem for degenerate singular points of germs of holomorphic vector fields in the complex plane. J. Dyn. Control Syst. 7(4), 553–599 (2001)CrossRefMATHGoogle Scholar

Ortiz-Bobadilla, L., Rosales-González, E., Voronin, S.M.: Rigidity theorems for generic holomorphic germs of dicritic foliations and vector fields in \(({\mathbb{C}}^2, 0)\). Mosc. Math. J. 5(1), 171–206 (2005)MathSciNetMATHGoogle Scholar

Ortiz-Bobadilla, L., Rosales-González, E., Voronin, S.M.: Analytic normal forms of germs of holomorphic dicritic foliations. Mosc. Math. J. 8(3), 521–545 (2008)MathSciNetMATHGoogle Scholar

Ortiz-Bobadilla, L., Rosales-González, E., Voronin, S.M.: Thom’s problem for degenerate singular points of holomorphic foliations in the plane. Mosc. Math. J. 12(4), 825–862 (2012)MathSciNetMATHGoogle Scholar

Ortiz-Bobadilla, L., Rosales-González, E., Voronin, S.M.: Formal and analytic normal forms of germs of holomorphic nondicritic foliations. J. Singul. 9, 168–192 (2014)MathSciNetMATHGoogle Scholar

Pérez-Marco, R.: Total convergence or general divergence in small divisors. Commun. Math. Phys. 223(3), 451–464 (2001)MathSciNetCrossRefMATHGoogle Scholar

Poincaré, H.: Note sur les propriétés des fonctions définies par des équations differéntielles. Journal de l’École Polytechnique 45, 13–26 (1878)Google Scholar

Ramis, J.P.: Confluence et resurgence. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36, 706–716 (1989)MATHGoogle Scholar

Shcherbakov, A.A.: Topological and analytical conjugacy of noncommutative groups of germs of conformal mappings. Trudy Sem. Petrovsk. 10, 170–196 (1984)MATHGoogle Scholar

Strozyna, E., Zoladek, H.: The analytic and formal normal forms for the nilpotent singularity. J. Differ. Equ. 179, 479–537 (2002)MathSciNetCrossRefMATHGoogle Scholar

Takens, F.: Normal forms for certain singularities of vector fields, Colloque International sur l’Analyse et la Topologie Différentielle, Ann. Inst. Fourier (Grenoble) 23(2), 163–195 (1973)Google Scholar

Voronin, S.M.: Analytic classification of germs of conformal mappings \(({\mathbb{C}}, 0) \rightarrow ({\mathbb{C}}, 0)\) with identity linear part. Funct. Anal. Appl. 15(1), 1–17 (1981)CrossRefMATHGoogle Scholar

Voronin, S.M.: Orbital analytic equivalence of degenerate singular points of holomorphic vector fields on the complex plane, Tr. Mat. Inst. Steklova 213, 30–49 (1997); Differ. Uravn. s Veshchestv. i Kompleks. Vrem., 35–55 (Russian)Google Scholar

Voronin, S.M., Grintchy, A.A.: An analytic classifications of saddle resonant singular points of holomorphic vector fields in the complex plane. J. Dyn. Control Syst. 2(1), 21–53 (1996)MathSciNetCrossRefMATHGoogle Scholar

Voronin, S.M., Meshcheryakova, Y.I.: Analytic classification of generic degenerate elementary singular points of gems of holomorphic vector fields on the complex plane, (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 46(1), 13–16 (2002). Translation in Russian Math. (Iz. VUZ). 46(1), 11–14 (2002)Google Scholar

Yoccoz, J.-C.: Linéarisation des germes de diffeómorphismes holomorphes de \(({\mathbb{C}}, 0)\). C. R. Acad. Sci. Paris Sér. I Math. 306(1), 55–58 (1988)MathSciNetMATHGoogle Scholar

Yoccoz, J.-C.: Théorème de Siegel, nombres de Bruno et polynômes quadratiques. Astérisque 231, 3–88 (1995)Google Scholar