Geodesic
Convergence of formal Dulac series satisfying an algebraic ordinary differential equation
Sbornik. Mathematics, Tome 210 (2019) no. 9, pp. 1207-1221 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A sufficient condition is proposed which ensures that a Dulac series that formally satisfies an algebraic ordinary differential equation (ODE) is convergent. Such formal solutions of algebraic ODEs are quite common: in particular, the Painlevé III, V and VI equations have formal solutions given by Dulac series; they are convergent in view of the sufficient condition presented. Bibliography: 13 titles.
Keywords: Dulac series
Mots-clés : algebraic ODE, formal solution, convergence. 
@article{SM_2019_210_9_a0,
     author = {R. R. Gontsov and I. V. Goryuchkina},
     title = {Convergence of formal {Dulac} series satisfying an algebraic ordinary differential equation},
     journal = {Sbornik. Mathematics},
     pages = {1207--1221},
     year = {2019},
     volume = {210},
     number = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2019_210_9_a0/}
}
TY  - JOUR
AU  - R. R. Gontsov
AU  - I. V. Goryuchkina
TI  - Convergence of formal Dulac series satisfying an algebraic ordinary differential equation
JO  - Sbornik. Mathematics
PY  - 2019
SP  - 1207
EP  - 1221
VL  - 210
IS  - 9
UR  - http://geodesic.mathdoc.fr/item/SM_2019_210_9_a0/
LA  - en
ID  - SM_2019_210_9_a0
ER  - 
%0 Journal Article
%A R. R. Gontsov
%A I. V. Goryuchkina
%T Convergence of formal Dulac series satisfying an algebraic ordinary differential equation
%J Sbornik. Mathematics
%D 2019
%P 1207-1221
%V 210
%N 9
%U http://geodesic.mathdoc.fr/item/SM_2019_210_9_a0/
%G en
%F SM_2019_210_9_a0
R. R. Gontsov; I. V. Goryuchkina. Convergence of formal Dulac series satisfying an algebraic ordinary differential equation. Sbornik. Mathematics, Tome 210 (2019) no. 9, pp. 1207-1221. http://geodesic.mathdoc.fr/item/SM_2019_210_9_a0/

[1] H. Dulac, “Sur les cycles limites”, Bull. Soc. Math. France, 51 (1923), 45–188 | DOI | MR | Zbl

[2] Yu. S. Il'yashenko, Finiteness theorems for limit cycles, Transl. from the Russian, Transl. Math. Monogr., 94, Amer. Math. Soc., Providence, RI, 1991, x+288 pp. | DOI | MR | Zbl

[3] Yu. S. Ilyashenko, “Finiteness theorems for limit cycles: a digest of the revised proof”, Izv. Math., 80:1 (2016), 50–112 | DOI | DOI | MR | Zbl

[4] A. D. Bruno, “Asymptotic behaviour and expansions of solutions of an ordinary differential equation”, Russian Math. Surveys, 59:3 (2004), 429–480 | DOI | DOI | MR | Zbl

[5] A. D. Bruno, I. V. Goryuchkina, “Asymptotic expansions of solutions of the sixth Painlevé equation”, Trans. Moscow Math. Soc., 2010, 1–104 | DOI | MR | Zbl

[6] S. Shimomura, “The sixth Painlevé transcendents and the associated Schlesinger equation”, Publ. Res. Inst. Math. Sci., 51:3 (2015), 417–463 | DOI | MR | Zbl

[7] S. Shimomura, “Logarithmic solutions of the fifth Painlevé equation near the origin”, Tokyo J. Math., 39:3 (2017), 797–825 | DOI | MR | Zbl

[8] B. Malgrange, “Sur le théorème de Maillet”, Asymptot. Anal., 2:1 (1989), 1–4 | MR | Zbl

[9] R. R. Gontsov, I. V. Goryuchkina, “On the convergence of generalized power series satisfying an algebraic ODE”, Asymptot. Anal., 93:4 (2015), 311–325 | DOI | MR | Zbl

[10] J. Cano, “On the series defined by differential equations, with an extension of the Puiseux polygon construction to these equations”, Analysis, 13:1-2 (1993), 103–119 | DOI | MR | Zbl

[11] Éd. Goursat, Cours d'analyse mathématique, v. I, Gauthier-Villars, Paris, 1902, 620 pp. | MR | Zbl

[12] A. D. Polyanin, V. F. Zaitsev, Handbook of exact solutions for ordinary differential equations, 2nd ed., Chapman Hall/CRC, Boca Raton, FL, 2003, xxvi+787 pp. | MR | Zbl | Zbl

[13] A. V. Gridnev, “Power expansions of solutions of the modified third Painlevé equation in a neighborhood of zero”, J. Math. Sci. (N.Y.), 145:5 (2007), 5180–5187 | DOI | MR | Zbl