Geodesic
Saddle connections
Sbornik. Mathematics, Tome 215 (2024) no. 11, pp. 1523-1548
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

It is shown that vector fields that are close to a fixed field with the same set of connections form a smooth Banach submanifold. A sufficient condition for the birth of saddle connections in a generic family is presented. The following result is proved: in a perturbation of a monodromic hyperbolic polycycle of $n$ connections in a generic family at least $n$ limit cycles can appear. Bibliography: 21 titles.
Keywords: separatrix connections, cyclicity.
Mots-clés : limit cycles, polycycles 
@article{SM_2024_215_11_a3,
     author = {A. V. Dukov},
     title = {Saddle connections},
     journal = {Sbornik. Mathematics},
     pages = {1523--1548},
     year = {2024},
     volume = {215},
     number = {11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2024_215_11_a3/}
}
TY  - JOUR
AU  - A. V. Dukov
TI  - Saddle connections
JO  - Sbornik. Mathematics
PY  - 2024
SP  - 1523
EP  - 1548
VL  - 215
IS  - 11
UR  - http://geodesic.mathdoc.fr/item/SM_2024_215_11_a3/
LA  - en
ID  - SM_2024_215_11_a3
ER  - 
%0 Journal Article
%A A. V. Dukov
%T Saddle connections
%J Sbornik. Mathematics
%D 2024
%P 1523-1548
%V 215
%N 11
%U http://geodesic.mathdoc.fr/item/SM_2024_215_11_a3/
%G en
%F SM_2024_215_11_a3
A. V. Dukov. Saddle connections. Sbornik. Mathematics, Tome 215 (2024) no. 11, pp. 1523-1548. http://geodesic.mathdoc.fr/item/SM_2024_215_11_a3/

[1] A. A. Andronov and E. A. Leontovich, “Generation of limit cycles from a separatrix forming a loop and from the separatrix of an equilibrium state of saddle-node type”, Amer. Math. Soc. Transl. Ser. 2, 33, Amer. Math. Soc., Providence, RI, 1963, 189–231 | DOI | MR | Zbl

[2] R. Abraham and J. Robbin, Transversal mappings and flows, W. A. Benjamin, Inc., New York–Amsterdam, 1967, x+161 pp. | MR | Zbl

[3] L. A. Cherkas, “The stability of singular cycles”, Differ. Equ., 4 (1972), 524–526 | MR | Zbl

[4] A. V. Dukov, Typical finite-parameter families of vector field on a two-dimensional sphere, Kandidat thesis, Moscow State University, Moscow, 2023, 84 pp. (Russian)

[5] A. V. Dukov, “Bifurcations of the ‘heart’ polycycle in generic 2-parameter families”, Trans. Moscow Math. Soc., 2018, 209–229 | DOI | MR | Zbl

[6] A. Dukov and Y. Ilyashenko, “Numeric invariants in semilocal bifurcations”, J. Fixed Point Theory Appl., 23:1 (2021), 3, 15 pp. | DOI | MR | Zbl

[7] Yu. Il'yashenko and S. Yakovenko, “Concerning the Hilbert sixteenth problem”, Concerning the Hilbert 16th problem, Amer. Math. Soc. Transl. Ser. 2, 165, Adv. Math. Sci., 23, Amer. Math. Soc., Providence, RI, 1995, 1–19 | DOI | MR | Zbl

[8] Yu. Ilyashenko, Yu. Kudryashov and I. Schurov, “Global bifurcations in the two-sphere: a new perspective”, Invent. Math., 213:2 (2018), 461–506 | DOI | MR | Zbl

[9] Yu. S. Il'yashenko and S. Yu. Yakovenko, “Finitely-smooth normal forms of local families of diffeomorphisms and vector fields”, Russian Math. Surveys, 46:1 (1991), 1–43 | DOI | MR | Zbl

[10] Yu. Ilyashenko and S. Yakovenko, “Finite cyclicity of elementary polycycles in generic families”, Concerning the Hilbert 16th problem, Amer. Math. Soc. Transl. Ser. 2, 165, Adv. Math. Sci., 23, Amer. Math. Soc., Providence, RI, 1995, 21–95 | DOI | MR | Zbl

[11] Yu. Ilyashenko and N. Solodovnikov, “Global bifurcations in generic one-parameter families with a separatrix loop on $S^2$”, Mosc. Math. J., 18:1 (2018), 93–115 | DOI | MR | Zbl

[12] Maoan Han, Yuhai Wu and Ping Bi, “Bifurcation of limit cycles near polycycles with $n$ vertices”, Chaos Solitons Fractals, 22:2 (2004), 383–394 | DOI | MR | Zbl

[13] 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

[14] Al Kelley, “The stable, center-stable, center, center-unstable, and unstable manifolds”: R. Abraham and J. Robbin, Transversal mappings and flows, W. A. Benjamin, Inc., New York–Amsterdam, 1967, 134–154 | MR | Zbl

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

[16] A. Mourtada, “Cyclicité finie des polycycles hyperboliques de champs de vecteurs du plan. Algorithme de finitude”, Ann. Inst. Fourier (Grenoble), 41:3 (1991), 719–753 | DOI | MR | Zbl

[17] A. Mourtada, Polycycles hyperboliques génériques à trois ou quatre sommets, Thése de doctorat, Dijon, 1990 http://www.theses.fr/1990DIJOS028

[18] J. W. Reyn, “Generation of limit cycles from separatrix polygons in the phase plane”, Geometrical approaches to differential equations (Scheveningen 1979), Lecture Notes in Math., 810, Springer, Berlin, 1980, 264–289 pp. | DOI | MR | Zbl

[19] V. Sh. Roitenberg, Nonlocal two-parameter bifurcations on surfaces, Ph.D. dissertation, Yaroslavl' State Technology University, Yaroslavl', 2000, 187 pp. (Russian)

[20] J. Sotomayor, “Generic one-parameter families of vector fields on two-dimensional manifolds”, Inst. Hautes Études Sci. Publ. Math., 43 (1974), 5–46 | DOI | MR | Zbl

[21] S. I. Trifonov, “Cyclicity of elementary polycycles of generic smooth vector fields”, Proc. Steklov Inst. Math., 213 (1996), 141–199 | MR | Zbl