Global phase portraits of Kukles differential systems with homogenous polynomial nonlinearities of degree 5 having a center
Keywords
Centers, Kukles, polynomial vector fields, phase portrait, Poincaré diskAbstract
We provide 22 different global phase portraits in the Poincaré disk of all centers of the so-called Kukles polynomial differential systems of the form $\dot{x} = -y$, $\dot{y} = x+Q_5(x,y)$, where $Q_5$ is a real homogeneous polynomial of degree 5 defined in $\mathbb{R}^2$.References
C.A. Buzzi, J. Llibre and 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.
R. Benterki and J. Llibre, Centers and their perturbation for the Kukles homogeneous polynomial differential systems of degree 4 symmetric with respect to the y-axis, preprint (2015).
F. Dumortier, J. Llibre and J.C. Artés, Qualitative Theory of Planar Differential Systems, Springer, 2006.
J. Giné, Conditions for the existence of a center for the Kukles homogeneous systems, Comput. Math. Appl. 43 (2002), 1261–1269.
J. Giné, J. Llibre and C. Valls, Centers for the Kukles homogeneous systems with odd degree, Bull. London Math. Soc. (2015), doi: 10.1112/blms/bdv005 (to appear).
J. Giné, J. Llibre and C. Valls, Centers for the Kukles homogeneous systems with even degree, preprint (2015).
J. Llibre and T. Salhi, Centers and their perturbation for the Kukles homogeneous polynomial differential systems of degree 4 symmetric with respect to the x-axis, preprint (2015).
K.E. Malkin, Criteria for the center for a certain differential equation, Volz. Mat. Sb. Vyp. 2 (1964), 87–91 (Russian).
L. Markus, Global structure of ordinary differential equations in the plane, Trans. Amer. Math. Soc. 76 (1954), 127–148.
D. Neumann, Classification of continuous flows on 2-manifold, Proc. Amer. Math. Soc. 48 (1975), 73–81.
E. P. Volokitin, V. V. Ivanov, Isochronicity and commutation of polynomial vector fields, Siberian Math. J. 40 (1999), 22–37.
N.I. Vulpe and K.S. Sibirskiı̆, Centro-affine invariant conditions for the existence of a center of a differential system with cubic nonlinearities, Dokl. Akad. Nauk SSSR 301 (1988), 1297–1301 (Russian); transl.: Soviet Math. Dokl. 38 (1989), 198-201.
H. Żołądek, The classification of reversible cubic systems with center, Topol. Methods Nonlinear Anal. 4 (1994), 79–136.
H. Żołądek, Remarks on: “The classification of reversible cubic systems with center” (Topol. Methods Nonlinear Anal. 4 (1994), 79–136), Topol. Methods Nonlinear Anal. 8 (1996), 335–342.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
