Geodesic
Bifurcations of the `heart' polycycle in generic 2-parameter families
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 79 (2018) no. 2, pp. 247-269 
A. V. Dukov. Bifurcations of the `heart' polycycle in generic 2-parameter families. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 79 (2018) no. 2, pp. 247-269. http://geodesic.mathdoc.fr/item/MMO_2018_79_2_a4/
@article{MMO_2018_79_2_a4,
     author = {A. V. Dukov},
     title = {Bifurcations of the `heart' polycycle in generic 2-parameter families},
     journal = {Trudy Moskovskogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {247--269},
     year = {2018},
     volume = {79},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MMO_2018_79_2_a4/}
}
TY  - JOUR
AU  - A. V. Dukov
TI  - Bifurcations of the `heart' polycycle in generic 2-parameter families
JO  - Trudy Moskovskogo matematičeskogo obŝestva
PY  - 2018
SP  - 247
EP  - 269
VL  - 79
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/MMO_2018_79_2_a4/
LA  - ru
ID  - MMO_2018_79_2_a4
ER  - 
%0 Journal Article
%A A. V. Dukov
%T Bifurcations of the `heart' polycycle in generic 2-parameter families
%J Trudy Moskovskogo matematičeskogo obŝestva
%D 2018
%P 247-269
%V 79
%N 2
%U http://geodesic.mathdoc.fr/item/MMO_2018_79_2_a4/
%G ru
%F MMO_2018_79_2_a4

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The paper concerns the ‘heart’ polycycle. We show that the set of vector fields containing a ‘heart’ polycycle form a Banach submanifold of codimension two in the space of smooth vector fields on a two-dimensional sphere. The bifurcation diagram of a generic family containing such a polycycle is constructed and surgery on the phase portrait is described.

[1] Ilyashenko Yu. S., Li Veigu, Nelokalnye bifurkatsii, Elektronnoe izdanie, MTsNMO, M., 2016, 327–335

[2] Malta I. P., Palis J., “Families of vector fields with finite modulus of stability”, Lecture Notes in Math., 898, Springer, Berlin–New York, 1981, 212–229 | DOI | MR | Zbl

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

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

[5] Kotova A., Stanzo V., “On few-parameter generic families of vector fields on the two-dimensional sphere”, Concerning the Hilbert 16th problem, AMS Transl. Ser. 2, 165, AMS, Providence, RI, 1995 ; Adv. Math. Sci., 23, 155–201 | MR | Zbl

[6] Andronov A. A., Pontryagin L. S., “Grubye sistemy”, DAN SSSR, 14:5 (1937), 247–250 | MR