×

Planar semi-quasi homogeneous polynomial differential systems with a given degree. (English) Zbl 1434.37012

Summary: This paper study the planar semi-quasi homogeneous polynomial differential systems (PSQHPDS), which can be regarded as a generalization of semi-homogeneous and of quasi-homogeneous systems. By using the algebraic skills, several important properties of PSQHPDS are derived and are employed to establish an algorithm for obtaining all the explicit expressions of PSQHPDS with a given degree. As an application of this algorithm, we research the center problem of quadratic and cubic PSQHPDS. The nonexistence of the quadratic center is proved and the canonical form of cubic center is found.

MSC:

37C10 Dynamics induced by flows and semiflows
34C25 Periodic solutions to ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Algaba, A., Freire, E., Gamero, E., García, C.: Monodromy, center-focus and integrability problems for quasi-homogeneous polynomial systems. Nonlinear Anal. 72, 1726-1736 (2010) · Zbl 1192.34035
[2] Algaba, A., García, C., Reyes, M.: Rational integrability of two dimensional quasi-homogeneous polynomial differential systems. Nonlinear Anal. 73, 1318-1327 (2010) · Zbl 1201.34050
[3] Algaba, A., García, C., Reyes, M.: Integratility of two dimensional quasi-homogeneous polynomial differential systems. Rocky Mt. J. Math. 41, 1-22 (2011) · Zbl 1213.37093
[4] Algaba, A., García, C., Teixeira, M.A.: Reversibility and quasi-homogeneous normal forms of vector fields. Nonlinear Anal. 73, 510-525 (2010) · Zbl 1202.34069
[5] Algaba, A., Fuentes, N., García, C.: Centers of quasi-homogeneous polynomial planar systems. Nonlinear Anal. Real world Appl. 13, 419-431 (2012) · Zbl 1238.34052
[6] Aziz, W., Llibre, J., Pantazi, C.: Centers of quasi-homogeneous polynomial differential equations of degree three. Adv. Math. 254, 233-250 (2014) · Zbl 1295.34042
[7] Cairó, L., Llibre, J.: Phase portraits of planar semi-homogeneous vector fields. I. Nonlinear Anal. 29, 783-811 (1997) · Zbl 0886.34026
[8] Cairó, L., Llibre, J.: Phase portraits of planar semi-homogeneous vector fields. II. Nonlinear Anal. 39, 351-363 (2000) · Zbl 0944.34022
[9] Cairó, L., Llibre, J.: Polynomial first integrals for weight-homogeneous planar polynomial differential systems of weight degree 3. J. Math. Anal. Appl. 331, 1284-1298 (2007) · Zbl 1124.34015
[10] Cairó, L., Llibre, J.: Phase portraits of planar semi-homogeneous vector fields. III. Qual. Theory Dyn. Syst. 10, 203-246 (2011) · Zbl 1270.34107
[11] Cima, A., Gasull, A., Maoñsas, F.: Limit cycles for vector fields with homogenous components. Appl. Math. 24, 281-287 (1997) · Zbl 0880.34032
[12] Cima, A., Llibre, J.: Algebraic and topological classification of the homogeneous cubic vector fields in the plane. J. Math. Anal. Appl. 147, 420-448 (1990) · Zbl 0711.34061
[13] Colak, I.E., Llibre, J., Valls, C.: Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields. Adv. Math. 259, 655-687 (2014) · Zbl 1303.34027
[14] Colak, I.E., Llibre, J., Valls, C.: Bifurcation diagrams for Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields. J. Differ. Equ. 258, 846-879 (2015) · Zbl 1309.34045
[15] Colak, I.E., Llibre, J., Valls, C.: Bifurcation diagrams for Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields. J. Differ. Equ. 262, 5518-5533 (2017) · Zbl 1369.34057
[16] Coll, B., Ferragut, A., Llibre, J.: Polynomial inverse integrating factors for quadratic differential systems. Nonlinear Anal. 73, 881-914 (2010) · Zbl 1203.34063
[17] Collins, C.B.: Algebraic classification of homogeneous polynomial vector fields in the plane. Jpn. J. Ind. Appl. Math. 13, 63-91 (1996) · Zbl 0855.34031
[18] Date, T., Lri, M.: Canonical forms of real homogeneous quadratic transformations. J. Math. Anal. Appl. 56, 650-682 (1976) · Zbl 0342.15009
[19] Dumortier, F., Llibre, J., Artés, J.C.: Qualititive Theory of Planar Differential Systems. Springer, New York (2006) · Zbl 1110.34002
[20] García, B., Llibre, J., Pérez del Río, J.S.: Planar quasi-homogeneous polynomial differential systems and their integrability. J. Differ. Equ. 255, 3185-3204 (2013) · Zbl 1336.34003
[21] García, I.A.: On the integrability of quasihomogeneous and related planar vector fields. Int. J. Bifurc. Chaos. 13, 995-1002 (2003) · Zbl 1077.34033
[22] Gavrilov, L., Giné, J., Grau, M.: On the cyclicity of weight-homogeneous centers. J. Differ. Equ. 246, 3126-3135 (2009) · Zbl 1182.34045
[23] Geng, F., Lian, H.: Bifurcation of limit cycles from a quasi-homogeneous degenerate center. Int. J. Bifurc. Chaos. 25, 1550007 (2015) · Zbl 1309.34066
[24] Ghose-Choudhury, A., Guha, P.: Monotonicity of the period function of the Liénard equation of second kind. Qual. Theory Dyn. Syst. 16, 609-621 (2017) · Zbl 1402.34043
[25] Giné, J., Grau, M., Llibre, J.: Limit cycles bifurcating from planar polynomial quasi-homogeneous centers. J. Differ. Equ. 259, 7135-7160 (2015) · Zbl 1332.34066
[26] Giné, J., Llibre, J., Valls, C.: Centers of weight-homogeneous polynomial vector fields on the plane. Proc. Am. Math. Soc. 145, 2539-2555 (2017) · Zbl 1380.37043
[27] Hu, Y.: On the integrability of quasihomogeneous systems and quasidegenerate infinite systems. Adv. Differ. Equ. 1, 1-10 (2007) · Zbl 1158.34003
[28] Kozlov, V.V., Furta, S.D.: Asymptotic Solutions of Strongly Nonlinear Systems of Differential Equations. Springer, New York (2013) · Zbl 1322.34003
[29] Li, W., Llibre, J., Yang, J., Zhang, Z.: Limit cycles bifurcating from the period annulus of quasihomogeneous centers. J. Dyn. Differ. Equ. 21, 133-152 (2009) · Zbl 1176.34038
[30] Liang, H., Huang, J., Zhao, Y.: Classification of global phase portraits of planar quartic quasihomogeneous polynomial differential systems. Nonlinear Dyn. 78, 1659-1681 (2014) · Zbl 1345.34068
[31] Liu, M.H., Guan, K.Y.: Reduction of quasi-homogeneous antonomous systems and reduced Kovalevskaya exponent (Chinese). Acta Math. Appl. Sin. 31, 729-743 (2008) · Zbl 1174.22005
[32] Llibre, J., Pérez del Río, J.S., Rodríguez, J.A.: Structural stability of planar homogeneous polynomial vector fields: applications to critical points and to infinity. J. Differ. Equ. 125, 490-520 (1996) · Zbl 0848.34037
[33] Llibre, J., Pérez del Río, J.S., Rodríguez, J.A.: Structural stability of planar semi-homogeneous polynomial vector fields. Applications to critical points and to infinity. Discrete Contin. Dyn. Syst. 6, 809-828 (2000) · Zbl 1011.37007
[34] Llibre, J., Pessoa, C.: On the centers of the weight-homogeneous polynomial vector fields on the plane. J. Math. Anal. Appl. 359, 722-730 (2009) · Zbl 1180.34030
[35] Llibre, J., Pessoa, C.: Phase portraits for quadratic homogeneous polynomial vector fields on \[S^2\] S2. Rend. Circ. Mat. Palermo 2(58), 361-406 (2009) · Zbl 1196.34040
[36] Moulin, O.J.: Liouvillian first integrals of homogeneous polynomial 3-dimensional vector fields. Colloq. Math. 70, 195-217 (1996) · Zbl 0851.35096
[37] Qiu, B., Liang, H.: Canonical forms of planar quasi-homogeneous coprime polynomial systems of degree 6. Int. J. Math. Anal. 9, 2539-2553 (2015)
[38] Qiu, B., Liang, H.: Classification of global phase portrait of planar quintic quasi-homogeneous coprime polynomial systems. Qual. Theory Dyn. Syst. 16, 417-451 (2017) · Zbl 1400.34045
[39] Sabatini, M.: On the period function of \[x^{\prime \prime }+f(x)x^{\prime 2}+g(x)=0\] x″+f(x)x′2+g(x)=0. J. Differ. Equ. 196, 151-168 (2004) · Zbl 1048.34068
[40] Shi, S., Zhu, W., Liu, B.: Non-existence of first integrals in a Laurent polynomial ring for general semi-quasihomogeneous systems. Z. Angew. Math. Phys. 57, 723-732 (2006) · Zbl 1112.34005
[41] Tang, Y., Zhang, X.: Center of planar quintic quasi-homogeneous polynomial differential systems. Discrete Contin. Dyn. Syst. 35, 2177-2191 (2015) · Zbl 1327.34054
[42] Xiong, Y., Han, M.: Planar quasi-homogeneous polynomial systems with a given weight degree. Discrete Contin. Dyn. Syst. 36, 4015-4025 (2016) · Zbl 1342.34049
[43] Yang, X.: Global phase-portraits of plane homogeneous polynomial vector fields and stability of the origin. Syst. Sci. Math. Sci. 10, 33-40 (1997) · Zbl 0883.34034
[44] Zhao, Y.: Limit cycles for planar semi-quasi-homogeneous polynomial vector fields. J. Math. Anal. Appl. 397, 276-284 (2013) · Zbl 1258.34059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.