Geodesic
Finiteness theorems for limit cycles: a digest of the revised proof
Izvestiya. Mathematics, Tome 80 (2016) no. 1, pp. 50-112 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This is the first paper in a series of two presenting a digest of the proof of the finiteness theorem for limit cycles of a planar polynomial vector field. At the same time we sketch the proof of the following two theorems: an analogous result for analytic vector fields, and a description of the asymptotics of the monodromy transformation for polycycles of such fields.
Keywords: elementary polycycles, superexact asymptotic series.
Mots-clés : limit cycles, functional cochains 
@article{IM2_2016_80_1_a3,
     author = {Yu. S. Ilyashenko},
     title = {Finiteness theorems for limit cycles: a digest of the revised proof},
     journal = {Izvestiya. Mathematics},
     pages = {50--112},
     year = {2016},
     volume = {80},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2016_80_1_a3/}
}
TY  - JOUR
AU  - Yu. S. Ilyashenko
TI  - Finiteness theorems for limit cycles: a digest of the revised proof
JO  - Izvestiya. Mathematics
PY  - 2016
SP  - 50
EP  - 112
VL  - 80
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/IM2_2016_80_1_a3/
LA  - en
ID  - IM2_2016_80_1_a3
ER  - 
%0 Journal Article
%A Yu. S. Ilyashenko
%T Finiteness theorems for limit cycles: a digest of the revised proof
%J Izvestiya. Mathematics
%D 2016
%P 50-112
%V 80
%N 1
%U http://geodesic.mathdoc.fr/item/IM2_2016_80_1_a3/
%G en
%F IM2_2016_80_1_a3
Yu. S. Ilyashenko. Finiteness theorems for limit cycles: a digest of the revised proof. Izvestiya. Mathematics, Tome 80 (2016) no. 1, pp. 50-112. http://geodesic.mathdoc.fr/item/IM2_2016_80_1_a3/

[1] 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. | MR | Zbl

[2] J. Écalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Actualites Math., Hermann, Paris, 1992, ii+340 pp. | MR | Zbl

[3] I. Bendixson, “Sur les courbes définies par des équations différentielles”, Acta Math., 24:1 (1901), 1–88 | DOI | MR | Zbl

[4] F. Dumortier, Singularities of vector fields, Monograf. Mat., 32, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1978, iv+191 pp. | MR

[5] A. Seidenberg, “Reduction of singularities of the differential equation $A\,dy=B\,dx$”, Amer. J. Math., 90:1 (1968), 248–269 | DOI | MR | Zbl

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

[7] Yu. S. Il'yashenko, “Limit cycles of polynomial vector fields with nondegenerate singular points on the real plane”, Funct. Anal. Appl., 18:3 (1984), 199–209 | DOI | MR | Zbl

[8] S. M. Voronin, “Analytic classification of germs of conformal mappings $(\mathbb{C},0)\to(\mathbb{C},0)$ with identity linear part”, Funct. Anal. Appl., 15:1 (1981), 1–13 | DOI | MR | Zbl

[9] Yu. S. Il'yashenko, “Dulac's memoir “On limit cycles” and related problems of the local theory of differential equations”, Russian Math. Surveys, 40:6 (1985), 1–49 | DOI | MR | Zbl

[10] Yu. S. Il'yashenko, “Separatrix lunes of analytic vector fields of the plane”, Mosc. Univ. Math. Bull., 41:4 (1986), 28–35 | MR | Zbl

[11] Yu. S. Il'yashenko, “Finiteness theorems for limit cycles”, Russian Math. Surveys, 45:2 (1990), 129–203 | DOI | MR | Zbl

[12] E. Ch. Titchmarsh, Teoriya funktsii, Gostekhizdat, M., 1951, 507 pp.; 2-Рμ РёР·Рґ., Наука, Рњ., 1980, 464 СЃ. ; E. C. Titchmarsh, The theory of functions, 2nd ed., Oxford Univ. Press, Oxford, 1939, x+454 СЃ. | MR | Zbl | MR | Zbl