On the centers of cubic polynomial differential systems with four invariant straight lines
Keywords
Cubic system, cubic polynomial differential systems, centers, invariant straight lineAbstract
Assume that a cubic polynomial differential system in the plane has four invariant straight lines in generic position, i.e., they are not parallel and no more than two straight lines intersect in a point. Then such a differential system only can have $0$, $1$ or $3$ centers.References
A.A. Andronov, E.A. Leontovich, I.I. Gordon and A.G. Maier, Qualitative Theory of Second-Order Dynamic Systems, Israel Program for Scientific Translations, Halsted Press (a division of Wiley), New York, 1973.
V.I. Arnold, A. Varchenko and S. Goussein-Zude, Singularités des Applications Différentiables, Mir, Moscou, 1982.
N.N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sbornik 30 (1952), 181–196; Amer. Math. Soc. Transl. 100 (1954), 1–19.
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.
I.E. Colak, J. Llibre and C. Valls, Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations 257 (2014), 1623–1661.
I.E. Colak, J. Llibre and C. Valls, Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields, Adv. Math. 259 (2014), 655–687.
I.E. Colak, J. Llibre and C. Valls, Bifurcation diagrams for Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations 258 (2015), 846–879.
I.E. Colak, J. Llibre and C. Valls, Bifurcation diagrams for Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations 262 (2017), 5518–5533.
C. Christopher, J. Llibre, C. Pantazi and X. Zhang, Darboux integrability and invariant algebraic curves for planar polynomial systems, J. Phys. A 35 (2002), 2457–2476.
H. Dulac, Détermination et integration d’une certaine classe d’équations différentielle ayant par point singulier un centre, Bull. Sci. Math. Sér. (2) 32 (1908), 230–252.
F. Dumortier, J. Llibre and J.C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer–Verlag, 2006.
W. Fulton, Algebraic Curves, Mathematics Lecture Note Series, W.A. Benjamin, 1974.
P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley–Interscience, New York, 1978.
D. Hilbert, Mathematische Probleme, Lecture, Second Internat. Congr. Math. (Paris, 1900), Nachr. Ges. Wiss. Göttingen Math. Phys. KL. (1900), 253–297; English transl.: Bull. Amer. Math. Soc. 8 (1902), 437–479; Bull. Amer. Math. Soc. (N.S.) 37 (2000), 407–436.
Yu. Ilyashenko, Centennial history of Hilbert’s 16th problem, Bull. Amer. Math. Soc. (N.S.) 39 (2002), 301–354.
W. Kapteyn, On the midpoints of integral curves of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland (1911), 1446–1457 (Dutch).
W. Kapteyn, New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk. 20 (1912), 1354–1365; 21, 27–33 (Dutch).
R.E. Kooij and C. Christopher, Algebraic invariant curves and the integrability of the polynomial systems, Appl. Math. Lett 6 (1993), no. 4, 51–53.
J. Li, Hilbert’s 16th problem and bifurcations of planar polynomial vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), 47–106.
J. Llibre and D. Xiao, On the configurations of centers of planar Hamiltonian Kolmogorov cubic polynomial differential systems, preprint, 2017.
K.E. Malkin, Criteria for the center for a certain differential equation, Volz. Mat. Sb. Vyp. 2 (1964), 87–91 (Russian).
I. Palasti, The maximal number of quadrilaterals bounded by general straight lines in a plane, Period. Math. Hungar. 6 (1975), no. 4, 323–341.
H. Poincaré, Mémoire sur les courbes définies par une équation differentielle, J. Maths. Pures Appl. 7 1881, 375–422.
D. Schlomiuk, Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc. 338 (1993), 799–841.
A. Suba and D. Cozma, Solution of the problem of the centre for cubic differential system with three invariant straight lines in generic position, Qual. Theory Dyn. Syst. 6 (2005), 45–58.
N. I. Vulpe, Affine–invariant conditions for the topological discrimination of quadratic systems with a center, Differential Equations 19 (1983), 273–280.
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.
Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Translations of Mathematical Monographs, vol. 101, Amer. Math. Soc., Providence, 1991.
H. Żołądek, Quadratic systems with center and their perturbations, J. Differential Equations 109 (1994), 223–273.
H. Żołądek, On a certain generalization of Bautin’s theorem, Nonlinearity 7 (1994), 273–279.
H. Żołądek, The classification of reversible cubic systems with center, Topol. Methods Nonlinear Anal. 4 (1994), 79–136.
H. Żołądek, On algebraic solutions of algebraic Pfaff equations, Studia Math. 114 (1995), no. 2, 117–126.
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
