Quadratic and сubic complex systems satisfying the Cauchy--Riemann conditions
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 12 (2015), pp. 45-63.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study plane quadratic and cubic differential systems satisfying the Cauсhy–Riemann conditions. We construct all global topologically equivalent phase portraits of the systems.
Keywords: quadratic systems, cubic systems, Darboux integrability, rational first integrals
Mots-clés : phase portraits.
@article{SEMR_2015_12_a59,
     author = {E. P. Volokitin and S. A. Treskov and V. M. Cheresiz},
     title = {Quadratic and {\cyrs}ubic complex systems satisfying the {Cauchy--Riemann} conditions},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {45--63},
     publisher = {mathdoc},
     volume = {12},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2015_12_a59/}
}
TY  - JOUR
AU  - E. P. Volokitin
AU  - S. A. Treskov
AU  - V. M. Cheresiz
TI  - Quadratic and сubic complex systems satisfying the Cauchy--Riemann conditions
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2015
SP  - 45
EP  - 63
VL  - 12
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2015_12_a59/
LA  - ru
ID  - SEMR_2015_12_a59
ER  - 
%0 Journal Article
%A E. P. Volokitin
%A S. A. Treskov
%A V. M. Cheresiz
%T Quadratic and сubic complex systems satisfying the Cauchy--Riemann conditions
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2015
%P 45-63
%V 12
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2015_12_a59/
%G ru
%F SEMR_2015_12_a59
E. P. Volokitin; S. A. Treskov; V. M. Cheresiz. Quadratic and сubic complex systems satisfying the Cauchy--Riemann conditions. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 12 (2015), pp. 45-63. http://geodesic.mathdoc.fr/item/SEMR_2015_12_a59/

[1] V. V. Amelkin, Chin Zan Dang, “Izokhronnost sistem Koshi–Rimana v sluchae fokusa”, Vestnik Akademii nauk Belarusi. Ser. fiz.-mat. nauk, 1 (1993), 28–31 | MR

[2] C. J. Christopher, J. Devlin, “Isochronous centers in planar polynomial systems”, SIAM J. Math. Anal., 28:1 (1997), 162–177 | DOI | MR | Zbl

[3] J. Gregor, “Dynamické systémy s regulární pravou stranou, I”, Pokroky matematiky, fyziky a astronomie, 3:2 (1958), 153–160 | Zbl

[4] N. I. Gavrilov, Metody teorii obyknovennykh differentsialnykh uravnenii, Vysshaya shkola, M., 1962 | MR

[5] N. A. Lukashevich, “Izokhronnost tsentra nekotorykh sistem differentsialnykh uravnenii”, Differents. uravneniya, 1:5 (1965), 295–302 | MR

[6] N. I. Pleshkan, “Novyi sposob issledovaniya na izokhronnost sistemy dvukh differentsialnykh uravnenii”, Differents. uravneniya, 5:6 (1969), 1083–1090 | MR | Zbl

[7] M. Villarini, “Regularity propeties of the period function near a centre of planar vector fields”, Nonlinear Anal., 19:8 (1992), 787–803 | DOI | MR | Zbl

[8] P. Mardesic, C. Rousseau, B. Toni, “Linearization of isochronous centers”, J. Differential Equations, 121:1 (1995), 67–108 | DOI | MR | Zbl

[9] E. P. Volokitin, V. M. Cheresiz, “Osobye tochki i pervye integraly golomorfnykh dinamicheskikh sistem”, Vestnik NGU. Ser.: Matematika, mekhanika, informatika, 13:2 (2013), 21–37 | Zbl

[10] C. J. Christopher, “Invariant algebraic curves and conditions for a centre”, Proc. R. Edinb. Sect A, 124:6 (1994), 1209–1229 | DOI | MR | Zbl

[11] E. P. Volokitin, “Integriruemost po Darbu polinomialnykh kompleksnykh sistem”, Sibirskie Elektronnye Matematicheskie Izvestiya, 10 (2013), 271–284 | MR

[12] E. P. Volokitin, “Neogranichennye resheniya polinomialnykh sistem Koshi–Rimana”, Sibirskie Elektronnye Matematicheskie Izvestiya, 11 (2014), 494–507

[13] E. A. Gonzáles Velasco, “Generic propeties of polynomial vector fields at infinity”, Trans. Amer. Math. Soc., 143 (1969), 201–222 | DOI | MR | Zbl

[14] N. N. Bautin, E. A. Leontovich, Metody i priemy kachestvennogo issledovaniya dinamicheskikh sistem na ploskosti, Nauka, M., 1990 | MR

[15] A. A. Andronov i dr., Kachestvennaya teoriya dinamicheskikh sistem na ploskosti, Nauka, M., 1966 | MR

[16] L. Markus, “Global structure of ordinary differential equations in the plane”, Trans. Amer. Math. Soc., 76 (1954), 127–148 | DOI | MR | Zbl

[17] D. A. Neumann, “Classification of continuous flows on 2-manifolds”, Proc. Amer. Math. Soc., 48:1 (1975), 73–81 | DOI | MR | Zbl

[18] J. Llibre, J. C. Medrado, “Darboux integrability and reversible quadratic vector fields”, Rocky Mountain J. Math., 35:6 (2005), 1999–2057 | DOI | MR | Zbl

[19] C. A. Buzzi, J. Llibre, J. C. R. Medrado, “Phase portraits of reversible linear differential systems with cubic homogeneous polynomial nonlinearities having a non-degenerate center at the origin”, Qual. Theory Dyn. Syst., 7 (2009), 369–403 | DOI | MR | Zbl

[20] G. Darboux, “Mémoire sur les équations différentielles algébrique du premier ordre et du premier degré (Mélanges)”, Bull. Sci. Math., 1878, 60–96 ; 123–144 ; 151–200 | Zbl

[21] I. A. Garcia, M. Grau, “A survey on the inverse integrating factor”, Qual. Theory Dyn. Syst., 9 (2010), 115–166 | DOI | MR | Zbl

[22] A. Ferragut, J. Llibre, “On the remarkable values of the rational first integrals of polynomial vector fields”, J. Differential Equations, 241 (2007), 399–417 | DOI | MR | Zbl

[23] H. Poincaré, “Sur líntegration algébrique des équations différentielles du premier ordre et du premier degré”, Rendiconti del Circolo Matematico di Palermo, 11 (1891), 193–239 ; 5 (1897), 161–191 | DOI | Zbl | DOI | Zbl

[24] R. Conti, “Centers of planar polynomial systems. A review”, Le Mathematiche, LIII:II (1998), 207–240 | MR | Zbl

[25] G. Polia, G. Segë, Zadachi i teoremy iz analiza, v. II, Nauka, M., 1978 | MR

[26] B. Coll, A. Ferragut, J. Llibre, “Phase portraits of the quadratic systems with a polynomial inverse integraiting factor”, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 2, 19:3 (2009), 765–783 | DOI | MR | Zbl