Geodesic
Multiplicities of limit cycles appearing after perturbations of hyperbolic polycycles
Sbornik. Mathematics, Tome 214 (2023) no. 2, pp. 226-245 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The problem of the multiplicity of limit cycles appearing after a perturbation of a hyperbolic polycycle with generic set of characteristic numbers is considered. In particular, it is proved that the multiplicity of any limit cycle appearing after a perturbation in a smooth finite-parameter family does not exceed the number of separatrix connections forming the polycycle. Bibliography: 10 titles.
Keywords: multiple fixed points.
Mots-clés : limit cycles, polycycles 
@article{SM_2023_214_2_a4,
     author = {A. V. Dukov},
     title = {Multiplicities of limit cycles appearing after perturbations of hyperbolic polycycles},
     journal = {Sbornik. Mathematics},
     pages = {226--245},
     year = {2023},
     volume = {214},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2023_214_2_a4/}
}
TY  - JOUR
AU  - A. V. Dukov
TI  - Multiplicities of limit cycles appearing after perturbations of hyperbolic polycycles
JO  - Sbornik. Mathematics
PY  - 2023
SP  - 226
EP  - 245
VL  - 214
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_2023_214_2_a4/
LA  - en
ID  - SM_2023_214_2_a4
ER  - 
%0 Journal Article
%A A. V. Dukov
%T Multiplicities of limit cycles appearing after perturbations of hyperbolic polycycles
%J Sbornik. Mathematics
%D 2023
%P 226-245
%V 214
%N 2
%U http://geodesic.mathdoc.fr/item/SM_2023_214_2_a4/
%G en
%F SM_2023_214_2_a4
A. V. Dukov. Multiplicities of limit cycles appearing after perturbations of hyperbolic polycycles. Sbornik. Mathematics, Tome 214 (2023) no. 2, pp. 226-245. http://geodesic.mathdoc.fr/item/SM_2023_214_2_a4/

[1] A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maĭer, Theory of bifurcations of dynamic systems on a plane, Nauka, Moscow, 1967, 487 pp. ; English transl., Halsted Press [John Wiley Sons], New York–Toronto; Israel Program for Scientific Translations, Jerusalem–London, 1973, xiv+482 pp. | MR | Zbl | MR

[2] F. Dumortier, R. Roussarie and C. Rousseau, “Elementary graphics of cyclicity 1 and 2”, Nonlinearity, 7:3 (1994), 1001–1043 | DOI | MR | Zbl

[3] T. M. Grozovskii, “Bifurcations of polycycles “apple” and “half-apple” in typical two-parameter families”, Differ. Uravn., 32:4 (1996), 458–469 ; English transl. in Differ. Equ., 32:4 (1996), 459–470 | MR | Zbl

[4] V. Sh. Roitenberg, Nonlocal two-parameter bifurcations of vector fields on surfaces, Kandidat dissertation, Yaroslavl' State Technical University, Yaroslavl', 2000 (Russian)

[5] S. I. Trifonov, “Cyclicity of elementary polycycles of generic smooth vector fields”, Differential equations with real and complex time, Tr. Mat. Inst. Steklova, 213, Nauka, Moscow, 1997, 152–212 ; English transl. in Proc. Steklov Inst. Math., 213 (1996), 141–199 | MR | Zbl

[6] Yu. Ilyashenko and S. Yakovenko, “Finite cyclicity of elementary polycycles in generic families”, Concerning the Hilbert sixteenth problem, Amer. Math. Soc. Transl. Ser. 2, 165, Adv. Math. Sci., 23, Amer. Math. Soc., Providence, RI, 1995, 21–65 | DOI | MR | Zbl

[7] V. Kaloshin, “The existential Hilbert 16-th problem and an estimate for cyclicity of elementary polycycles”, Invent. Math., 151:3 (2003), 451–512 | DOI | MR | Zbl

[8] P. I. Kaleda and I. V. Shchurov, “Cyclicity of elementary polycycles with fixed number of singular points in generic $k$-parameter families”, Algebra i Analiz, 22:4 (2010), 57–75 ; English transl. in St. Petersburg Math. J., 22:4 (2011), 557–571 | MR | Zbl | DOI

[9] D. Eisenbud, Commutative algebra. With a view toward algebraic geometry, Grad. Texts in Math., 150, Springer-Verlag, New York, 1995, xvi+785 pp. | DOI | MR | Zbl

[10] V. V. Prasolov, Polynomials, 2nd ed., Moscow Center for Continuous Mathematical Education, Moscow, 2001, 335 pp.; English transl., Algorithms Comput. Math., 11, Springer-Verlag, Berlin, 2004, xiv+301 pp. | DOI | MR | Zbl