Geodesic
Isochronous centers and foci of two-dimensional holomorphic differential systems
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2024), pp. 3-11 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Two-dimensional autonomous systems of differential equations of the form $ \dot{x}=-y-P(x,y), \dot{y}=x+Q(x,y), $ where $P$ and $Q$ are holomorphic functions of order greater than or equal two are considered. In this work, necessary and sufficient conditions for the existence of an isochronous center or focus are obtained. These conditions are formulated in terms of commuting differential systems and some normal form.
Keywords: two-dimensional holomorphic differential system, commuting differential system, normal form.
Mots-clés : isochronous center, isochronous focus 
@article{IVM_2024_11_a0,
     author = {V. V. Amel'kin and V. Yu. Tyshchenko},
     title = {Isochronous centers and foci of two-dimensional holomorphic differential systems},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {3--11},
     year = {2024},
     number = {11},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2024_11_a0/}
}
TY  - JOUR
AU  - V. V. Amel'kin
AU  - V. Yu. Tyshchenko
TI  - Isochronous centers and foci of two-dimensional holomorphic differential systems
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2024
SP  - 3
EP  - 11
IS  - 11
UR  - http://geodesic.mathdoc.fr/item/IVM_2024_11_a0/
LA  - ru
ID  - IVM_2024_11_a0
ER  - 
%0 Journal Article
%A V. V. Amel'kin
%A V. Yu. Tyshchenko
%T Isochronous centers and foci of two-dimensional holomorphic differential systems
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2024
%P 3-11
%N 11
%U http://geodesic.mathdoc.fr/item/IVM_2024_11_a0/
%G ru
%F IVM_2024_11_a0
V. V. Amel'kin; V. Yu. Tyshchenko. Isochronous centers and foci of two-dimensional holomorphic differential systems. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2024), pp. 3-11. http://geodesic.mathdoc.fr/item/IVM_2024_11_a0/

[1] Algaba A., Reyes M., “Characterizing isochronous points and computing isochronous sections”, J. Math. Anal. Appl., 355:2 (2009), 564–576 | DOI | MR | Zbl

[2] Amelkin V.V., Lukashevich N.A., Sadovskii A.P., Nelineinye kolebaniya v sistemakh vtorogo poryadka, BGU, Minsk, 1982

[3] Van D., Li Ch., Chou Sh.-N., Normalnye formy i bifurkatsii vektornykh polei na ploskosti, MTsNMO, M., 2005

[4] Bryuno A.D., “Analiticheskaya forma differentsialnykh uravnenii”, Tr. Moskovsk. matem. ob-va, 25, 1971, 119–262 | Zbl

[5] Tokarev S.P., “Gladkaya ekvivalentnost sistem differentsialnykh uravnenii na ploskosti v sluchae fokusa”, Differents. uravneniya, 13:5 (1977), 892–897 | MR | Zbl

[6] Abdullaev N., “Ob izokhronnosti pri nelineinykh kolebaniyakh”, Tr. Tadzhiks. uchitelsk. in-ta im. S.S. Aini, 2 (1954), 71–78

[7] Volokitin E.P., Ivanov B.B., “Izokhronnost i kommutiruemost polinomialnykh vektornykh polei”, Sib. matem. zhurn., 40:1 (1999), 30–48 | MR | Zbl

[8] Ladis N.N., “Kommutiruyuschie vektornye polya i izokhronnost”, Vestn. Belorussk. gos. un-ta. Ser. 1, 1976, no. 1, 21–24 | MR

[9] Villarini M., “Regularity properties of the period function near a center of a planar vector field”, Nonlinear Anal., 8 (1992), 787–803 | DOI | MR | Zbl

[10] Sabatini M., “Characterizing isochronous centers by Lie brackets”, Diff. Equat. Dynam. Syst., 5:1 (1997), 91–99 | MR | Zbl

[11] Llibre J., Ramirez R., Ramirez V., Sadovskaia N., “Centers and uniform isochronous centers of planar polynomial differential systems”, J. Dyn. Diff. Equat., 30:3 (2018), 1295–1310 | DOI | MR | Zbl

[12] Algaba A., Freire E., Gamero E., “Isochronicity via normal form”, Qual. Theory Dynam. Syst., 1:5 (2000), 133–156 | DOI | MR | Zbl

[13] Giné J., “Isochronous foci for analytic differential systems”, Int. J. Bifurcation Chaos., 13:6 (2003), 1617–1623 | DOI | MR | Zbl

[14] Giné J., Grau M., “Characterization of isochronous foci for planar analytic differential systems”, Proc. Royal Soc. Edinburgh. Sect. A.: Math., 135:5 (2005), 985–998 | DOI | MR | Zbl

[15] Arnold V.I., Dopolnitelnye glavy teorii obyknovennykh differentsialnykh uravnenii, Nauka, M., 1978

[16] Bibikov Yu.N., Kurs obyknovennykh differentsialnykh uravnenii, Vyssh. shk., M., 1991 | MR