Algebraic limit cycles of planar cubic systems
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 2045-2054.

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

We study algebraic limit cycles of differential systems of the form $\dot x= x+P_3(x,y), \ \dot y=y+Q_3(x,y)$ where $P_3(x,y)$ and $Q_3(x,y)$ are homogeneous cubic polynomials.
Keywords: polynomial systems
Mots-clés : algebraic limit cycles, non-algebraic limit cycles, phase portraits.
@article{SEMR_2020_17_a110,
     author = {E. P. Volokitin and V. M. Cheresiz},
     title = {Algebraic limit cycles of planar cubic systems},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {2045--2054},
     publisher = {mathdoc},
     volume = {17},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a110/}
}
TY  - JOUR
AU  - E. P. Volokitin
AU  - V. M. Cheresiz
TI  - Algebraic limit cycles of planar cubic systems
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2020
SP  - 2045
EP  - 2054
VL  - 17
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2020_17_a110/
LA  - en
ID  - SEMR_2020_17_a110
ER  - 
%0 Journal Article
%A E. P. Volokitin
%A V. M. Cheresiz
%T Algebraic limit cycles of planar cubic systems
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2020
%P 2045-2054
%V 17
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2020_17_a110/
%G en
%F SEMR_2020_17_a110
E. P. Volokitin; V. M. Cheresiz. Algebraic limit cycles of planar cubic systems. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 2045-2054. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a110/

[1] G. Darboux, “Mémoire sur les équations différentielles algébrique du premier ordre et du premier degré (Mélanges)”, Darboux Bull. (2), II (1878), 60–96 ; 123–144 ; 151–200 | Zbl

[2] H. Poincaré, “Sur lìntégration algébrique des équations différentielles du premier ordre et du premier degré”, Rend. Circ. Mat. Palermo, 5 (1891), 161–191 | DOI | Zbl

[3] J. Llibre, X. Zhang, “A survay on algebraic end explicit non-algebraic limit cycles in planar differential systems”, Exposiones Mathematicae (to appear) (Available online 15 June 2020) | DOI | MR

[4] A. Bendjeddou, J. Llibre, T. Salhi, “Dynamics of the polynomial differential systems with homogeneous nonlinearities and a star node”, J. Diff. Equ., 254:8 (2013), 3530–3537 | DOI | MR | Zbl

[5] E.P. Volokitin, V.M, Cheresiz, “Qualitative investigation of plane polynomial Darboux-type systems”, Sib. Electron. Mat. Izv., 13 (2016), 1170–1186 | MR | Zbl

[6] J. Giné, M. Grau, “Coexistence of algebraic and non-algebraic limit cycles, explicity given, using Riccati equations”, Nonlinearity, 19:8 (2006), 1939–1950 | DOI | MR | Zbl

[7] A. Cima, J. Llibre, “Algebraic and topological classification of the homogeneous cubic vector fields in the plane”, J. Math. Anal. Appl., 147:2 (1990), 420–448 | DOI | MR | Zbl

[8] Y.-c. Liu, “On differential equation with algebraic limit cycle of second degree $dy/dx=(a_{10}x+a_{01}y+a_{30}x^3+a_{21}x^2y+a_{12}xy^2+a_{03}y^3)/ (b_{10}x+b_{01}y+b_{30}x^3+b_{21}x^2y+b_{12}xy^2+b_{03}y^3)$”, Advancement in Math., 4 (1958), 143–149 | MR

[9] P.S. Belevec, R.T. Valeeva, “A class of dynamic systems with an algebraic limit cycle”, Kuibyshev. Gos. Ped. Inst. Nauchn. Trudy, 232 (1979), 5–8 | MR

[10] B.Q. Shen, “Existence of limit cycles of a system $(E_3^2)$ with elliptic solution”, Chin. Ann. Math. A, 14:6 (1993), 698–706 | MR

[11] R. Benterki, J. Llibre, “Polynomial differential systems with explicit non-algebraic limit cycles”, Electron. J. Differ. Equ., 2012 (2012), 78 | DOI | MR | Zbl