# American Institute of Mathematical Sciences

July  2016, 36(7): 4015-4025. doi: 10.3934/dcds.2016.36.4015

## Planar quasi-homogeneous polynomial systems with a given weight degree

 1 School of Mathematics and Statistics, Nanjing University of Information Science & Technology, Nanjing 210044, China 2 Department of Mathematics, Shanghai Normal University, Shanghai 200234

Received  January 2015 Revised  January 2016 Published  March 2016

In this paper, we investigate a class of quasi-homogeneous polynomial systems with a given weight degree. Firstly, by some analytical skills, several properties about this kind of systems are derived and an algorithm can be established to obtain all possible explicit systems for a given weight degree. Then, we focus on center problems for such systems and provide some necessary conditions for the existence of centers. Finally, for a specific quasi-homogeneous polynomial system, we characterize its center and prove that the center is not isochronous.
Citation: Yanqin Xiong, Maoan Han. Planar quasi-homogeneous polynomial systems with a given weight degree. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 4015-4025. doi: 10.3934/dcds.2016.36.4015
##### References:
 [1] W. Aziz, J. Llibre and C. Pantazi, Centers of quasi-homogeneous polynomial differential equations of degree three, Advances in Mathematics, 254 (2014), 233-250. doi: 10.1016/j.aim.2013.12.006. [2] A. Cima and J. Llibre, Algebraic and topological classification of the homogeneous cubic systems in the plane, J. Math. Anal. Appl., 147 (1990), 420-448. doi: 10.1016/0022-247X(90)90359-N. [3] T. Date and M. Lai, Canonical forms of real homogeneous quadratic transformations, J. Math. Anal. Appl., 56 (1976), 650-682. doi: 10.1016/0022-247X(76)90031-7. [4] T. Date, Classification and analysis of two-dimensional homogeneous quadratic differential equations systems, J. Differential Equations, 32 (1979), 311-334. doi: 10.1016/0022-0396(79)90037-8. [5] F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theorey of Planar Polynomial Systems, Springer, 2006. [6] B. García, J. Llibre and J. S. Pérea del Río, Planar quasi-homogeneous polynomial differential systems and their integrability, J. Differential Equations, 255 (2013), 3185-3204. doi: 10.1016/j.jde.2013.07.032. [7] L. Gavrilov, J. Giné and M. Grau, On the cyclicity of weight-homogeneous centers, J. Differential Equations, 246 (2009), 3126-3135. doi: 10.1016/j.jde.2009.02.010. [8] J. Giné, M. Grau and J. Llibre, Polynomial and rational first integrals for planar quasi-homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst., 33 (2013), 4531-4547. doi: 10.3934/dcds.2013.33.4531. [9] J. Llibre and X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems, Nonlinearity, 15 (2002), 1269-1280. doi: 10.1088/0951-7715/15/4/313. [10] J. Llibre and C. Pessoa, On the centers of the weight-homogeneous polynomial vector fields on the plane, J. Math. Anal. Appl., 359 (2009), 722-730. doi: 10.1016/j.jmaa.2009.06.036. [11] W. Li, J. Llibre, J. Yang and Z. Zhang, Limit cycles bifurcating from the period annulus of quasi-homogeneous centers, J. Dyn. Diff. Equat., 21 (2009), 133-152. doi: 10.1007/s10884-008-9126-1. [12] H. Liang, J. Huang and Y. Zhao, Classification of global phase portraits of planar quartic quasi-homogeneous polynomial differential systems, Nonlinear Dynamics, 78 (2014), 1659-1681. doi: 10.1007/s11071-014-1541-8. [13] P. Mardešić, C. Rousseau and B. Toni, Linearization of isochronous centers, J. Differential Equations, 121 (1995), 67-108. doi: 10.1006/jdeq.1995.1122. [14] Y. Zhao, Limit cycles for planar semi-quasi-homogeneous polynomial vector fields, J. Math. Anal. Appl., 397 (2013), 276-284. doi: 10.1016/j.jmaa.2012.07.060.

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##### References:
 [1] W. Aziz, J. Llibre and C. Pantazi, Centers of quasi-homogeneous polynomial differential equations of degree three, Advances in Mathematics, 254 (2014), 233-250. doi: 10.1016/j.aim.2013.12.006. [2] A. Cima and J. Llibre, Algebraic and topological classification of the homogeneous cubic systems in the plane, J. Math. Anal. Appl., 147 (1990), 420-448. doi: 10.1016/0022-247X(90)90359-N. [3] T. Date and M. Lai, Canonical forms of real homogeneous quadratic transformations, J. Math. Anal. Appl., 56 (1976), 650-682. doi: 10.1016/0022-247X(76)90031-7. [4] T. Date, Classification and analysis of two-dimensional homogeneous quadratic differential equations systems, J. Differential Equations, 32 (1979), 311-334. doi: 10.1016/0022-0396(79)90037-8. [5] F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theorey of Planar Polynomial Systems, Springer, 2006. [6] B. García, J. Llibre and J. S. Pérea del Río, Planar quasi-homogeneous polynomial differential systems and their integrability, J. Differential Equations, 255 (2013), 3185-3204. doi: 10.1016/j.jde.2013.07.032. [7] L. Gavrilov, J. Giné and M. Grau, On the cyclicity of weight-homogeneous centers, J. Differential Equations, 246 (2009), 3126-3135. doi: 10.1016/j.jde.2009.02.010. [8] J. Giné, M. Grau and J. Llibre, Polynomial and rational first integrals for planar quasi-homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst., 33 (2013), 4531-4547. doi: 10.3934/dcds.2013.33.4531. [9] J. Llibre and X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems, Nonlinearity, 15 (2002), 1269-1280. doi: 10.1088/0951-7715/15/4/313. [10] J. Llibre and C. Pessoa, On the centers of the weight-homogeneous polynomial vector fields on the plane, J. Math. Anal. Appl., 359 (2009), 722-730. doi: 10.1016/j.jmaa.2009.06.036. [11] W. Li, J. Llibre, J. Yang and Z. Zhang, Limit cycles bifurcating from the period annulus of quasi-homogeneous centers, J. Dyn. Diff. Equat., 21 (2009), 133-152. doi: 10.1007/s10884-008-9126-1. [12] H. Liang, J. Huang and Y. Zhao, Classification of global phase portraits of planar quartic quasi-homogeneous polynomial differential systems, Nonlinear Dynamics, 78 (2014), 1659-1681. doi: 10.1007/s11071-014-1541-8. [13] P. Mardešić, C. Rousseau and B. Toni, Linearization of isochronous centers, J. Differential Equations, 121 (1995), 67-108. doi: 10.1006/jdeq.1995.1122. [14] Y. Zhao, Limit cycles for planar semi-quasi-homogeneous polynomial vector fields, J. Math. Anal. Appl., 397 (2013), 276-284. doi: 10.1016/j.jmaa.2012.07.060.
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