Geodesic
Centers and limit cycles of generalized Kukles polynomial differential systems: phase portraits and limit cycles
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 13 (2020) no. 4, pp. 387-397.

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

In this work, we give the seven global phase portraits in the Poincaré disc of the Kukles differential system given by \begin{equation*} \begin{array}{l} \dot{x} = -y,\\ \dot{y}= x + a x^8 + b x^4 y^4 + cy^8, \end{array} \end{equation*} where $x, y \in \mathbb{R}$ and $a, b, c \in \mathbb{R}$ with $a^2 + b^2 + c^2 \neq 0$. Moreover, we perturb these system inside all classes of polynomials of eight degrees, then we use the averaging theory up sixth order to study the number of limit cycles which can bifurcate from the origin of coordinates of the Kukles differential system.
Keywords: limit cycle, generalized Kukles differential system, averaging method
Mots-clés : phase portrait. 
@article{JSFU_2020_13_4_a0,
     author = {Ahlam Belfar and Rebiha Benterki},
     title = {Centers and limit cycles of generalized {Kukles} polynomial differential systems: phase portraits and limit cycles},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {387--397},
     publisher = {mathdoc},
     volume = {13},
     number = {4},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2020_13_4_a0/}
}
TY  - JOUR
AU  - Ahlam Belfar
AU  - Rebiha Benterki
TI  - Centers and limit cycles of generalized Kukles polynomial differential systems: phase portraits and limit cycles
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2020
SP  - 387
EP  - 397
VL  - 13
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2020_13_4_a0/
LA  - en
ID  - JSFU_2020_13_4_a0
ER  - 
%0 Journal Article
%A Ahlam Belfar
%A Rebiha Benterki
%T Centers and limit cycles of generalized Kukles polynomial differential systems: phase portraits and limit cycles
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2020
%P 387-397
%V 13
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2020_13_4_a0/
%G en
%F JSFU_2020_13_4_a0
Ahlam Belfar; Rebiha Benterki. Centers and limit cycles of generalized Kukles polynomial differential systems: phase portraits and limit cycles. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 13 (2020) no. 4, pp. 387-397. http://geodesic.mathdoc.fr/item/JSFU_2020_13_4_a0/

[1] R. Benterki, J. Llibre, “Centers and limit cycles of polynomial differential systems of degree $4$ via averaging theory”, J. Computational and Appl. Math., 313 (2017), 273–283

[2] R. Benterki, J. Llibre, “The centers and their cyclicity for a class of polynomial differential systems of degree 7 via averaging theory”, J. Computational and Appl. Math., 368 (2020), 112456

[3] C.A. Buzzi, J. Llibre, J.C. 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

[4] F. Dumortier, J. Llibre, J.C. Artés, Qualitative theory of planar differential systems, Universitext, Springer-Verlag, 2006

[5] J. Giné, “Conditions for the existence of a center for the Kukles homogenenous systems”, Comput. Math. Appl., 43 (2002), 1261–1269

[6] J. Giné, J. LLibre, C. Valls, “Centers for the Kukles homogeneous systems with odd degree”, Bull. London Math. Soc., 47 (2015), 315–324

[7] J. Giné, J. LLibre, C. Valls, “Centers for the Kukles homogeneous systems with even degree”, J. Appl. Anal. Comp., 7 (2017) | DOI

[8] J. Llibre, D.D. Novaes, M.A. Teixeira, Nonlinearity, 27 (2014), 563–583 | DOI

[9] J. Llibre, M.F. da Silva, Int. J. of Bifurcation and Chaos, 26:3 (2016), 1650044 | DOI

[10] J. Llibre, M.F. da Silva, “Global phase portraits of Kukles differential systems with homogenous polynomial nonlinearities of degree $5$ having a center”, Topological Methods in Nonlinear Analysis, 48 (2016), 257–282

[11] K.E. Malkin, “Criteria for the center for a certain differential equation”, Volzhskii Matematicheskii Sbornik, 2 (1964), 87–91 (in Russian)

[12] E.P. Volokitin, V.V. Ivanov, “Isochronicity and Commutation of polynomial vector fields”, Siberian Mathematical Journal, 40 (1999), 23–38

[13] N.I. Vulpe, K.S. Sibirskii, “Centro–affine invariant conditions for the existence of a center of a differential system with cubic nonlinearities”, Soviet Math. Dokl., 38 (1989), 198–201

[14] H. Żoła̧dek, “The classification of reversible cubic systems with center”, Topol. Methods Nonlinear Anal., 4 (1994), 79–136

[15] H. Żoła̧dek, “Remarks on: The classification of reversible cubic systems with center”, Topol. Methods Nonlinear Anal., 8 (1996), 335–342