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January  2022, 27(1): 421-441. doi: 10.3934/dcdsb.2021049

Limit cycles and global dynamic of planar cubic semi-quasi-homogeneous systems

1. 

School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou 510665, China

2. 

School of Mathematical Sciences and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: Haihua Liang

Received  September 2020 Revised  December 2020 Published  January 2022 Early access  February 2021

Fund Project: The second author is supported by the NNSF of China grant 11771101, by the major research program of colleges and universities in Guangdong Province grant 2017KZDXM054, and by the Science and Technology Program of Guangzhou of China grant 201805010001. The third author is partially supported by NNSF of China grant 11671254, 11871334 and 12071284

Denote by CH, CSH, CQH, and CSQH the planar cubic homogeneous, cubic semi-homogeneous, cubic quasi-homogeneous and cubic semi-quasi-homogeneous differential systems, respectively. The problems on limit cycles and global dynamics of these systems have been solved for CH, and partially for CSH. This paper studies the same problems for CQH and CSQH. We prove that CQH have no limit cycles and CSQH can have at most one limit cycle with the limit cycle realizable. Moreover, we classify all the global phase portraits of CSQH.

Citation: Zecen He, Haihua Liang, Xiang Zhang. Limit cycles and global dynamic of planar cubic semi-quasi-homogeneous systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 421-441. doi: 10.3934/dcdsb.2021049
References:
[1]

A. AlgabaC. Garcia and M. Reyes, Integrability of two dimensional quasi-homogeneous polynomial differential systems, Rocky Mountain J. Math., 41 (2011), 1-22.  doi: 10.1216/RMJ-2011-41-1-1.

[2]

W. AzizJ. Llibre and C. Pantazi, Centers of quasi-homogeneous polynomial differential equations of degree three, Adv. Math., 254 (2014), 233-250.  doi: 10.1016/j.aim.2013.12.006.

[3]

L. Cairó and J. Llibre, Phase portraits of planar semi-homogeneous vector fields (Ⅱ), Nonlinear Anal., 39 (2000), 351-363. 

[4]

L. Cairó and J. Llibre, Phase portraits of planar semi-homogeneous vector fields (Ⅲ), Qual. Theory Dyn. Syst., 10 (2011), 203-246.  doi: 10.1007/s12346-011-0052-y.

[5]

L. Cairó and J. Llibre, Polynomial first integrals for weight-homogeneous planar polynomial differential systems of weight degree 3, J. Math. Anal. Appl., 331 (2007), 1284-1298.  doi: 10.1016/j.jmaa.2006.09.066.

[6]

A. CimaA. Gasull and F. Ma$\tilde{n}$osas, Cyclicity of a family of vector fields, J. Math. Anal. Appl., 196 (1995), 921-937.  doi: 10.1006/jmaa.1995.1451.

[7]

A. CimaA. Gasull and F. Ma$\tilde{n}$osas, Limit cycles for vector fields with homogeneous components, Appl. Math. (Warsaw), 24 (1997), 281-287.  doi: 10.4064/am-24-3-281-287.

[8]

A. Cima and J. Llibre, Algebraic and topological classification of the homogeneous cubic vector fields in the plane, J. Math. Anal. Appl., 147 (1990), 420-448.  doi: 10.1016/0022-247X(90)90359-N.

[9]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006. doi: 10.1007/978-3-540-32902-2.

[10]

B. GarcíaJ. Llibre and J. S. Pérez 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.

[11]

G. HuangG. Feng and X. Zhang, A global topological structure of a class of cubic quasi-homogeneous vector fields, Acta Math. Sci. A, 34 (2014), 419-425. 

[12]

H. LiangJ. Huang and Y. Zhao, Classification of global phase portraits of planar quartic quasi-homogeneous polynomial differential systems, Nonlinear Dynam., 78 (2014), 1659-1681.  doi: 10.1007/s11071-014-1541-8.

[13]

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.

[14]

B. Qiu and H. Liang, Classification of global phase portrait of planar quintic quasi-homogeneous coprime polynomial systems, Qual. Theory Dyn. Syst., 16 (2017), 417-451.  doi: 10.1007/s12346-016-0199-7.

[15]

Y. TangL. Wang and X. Zhang, Center of planar quintic quasi-homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst., 35 (2015), 2177-2191.  doi: 10.3934/dcds.2015.35.2177.

[16]

Y. Tian and H. Liang, Planar semi-quasi homogeneous polynomial differential systems with a given degree, Qual. Theory Dyn. Syst., 18 (2019), 841-871.  doi: 10.1007/s12346-019-00316-w.

[17]

Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Volume 101 of Trans. of Mathematical Monographs, Am. Math. Soc. Providence, RI, 1992.

[18]

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.

show all references

References:
[1]

A. AlgabaC. Garcia and M. Reyes, Integrability of two dimensional quasi-homogeneous polynomial differential systems, Rocky Mountain J. Math., 41 (2011), 1-22.  doi: 10.1216/RMJ-2011-41-1-1.

[2]

W. AzizJ. Llibre and C. Pantazi, Centers of quasi-homogeneous polynomial differential equations of degree three, Adv. Math., 254 (2014), 233-250.  doi: 10.1016/j.aim.2013.12.006.

[3]

L. Cairó and J. Llibre, Phase portraits of planar semi-homogeneous vector fields (Ⅱ), Nonlinear Anal., 39 (2000), 351-363. 

[4]

L. Cairó and J. Llibre, Phase portraits of planar semi-homogeneous vector fields (Ⅲ), Qual. Theory Dyn. Syst., 10 (2011), 203-246.  doi: 10.1007/s12346-011-0052-y.

[5]

L. Cairó and J. Llibre, Polynomial first integrals for weight-homogeneous planar polynomial differential systems of weight degree 3, J. Math. Anal. Appl., 331 (2007), 1284-1298.  doi: 10.1016/j.jmaa.2006.09.066.

[6]

A. CimaA. Gasull and F. Ma$\tilde{n}$osas, Cyclicity of a family of vector fields, J. Math. Anal. Appl., 196 (1995), 921-937.  doi: 10.1006/jmaa.1995.1451.

[7]

A. CimaA. Gasull and F. Ma$\tilde{n}$osas, Limit cycles for vector fields with homogeneous components, Appl. Math. (Warsaw), 24 (1997), 281-287.  doi: 10.4064/am-24-3-281-287.

[8]

A. Cima and J. Llibre, Algebraic and topological classification of the homogeneous cubic vector fields in the plane, J. Math. Anal. Appl., 147 (1990), 420-448.  doi: 10.1016/0022-247X(90)90359-N.

[9]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006. doi: 10.1007/978-3-540-32902-2.

[10]

B. GarcíaJ. Llibre and J. S. Pérez 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.

[11]

G. HuangG. Feng and X. Zhang, A global topological structure of a class of cubic quasi-homogeneous vector fields, Acta Math. Sci. A, 34 (2014), 419-425. 

[12]

H. LiangJ. Huang and Y. Zhao, Classification of global phase portraits of planar quartic quasi-homogeneous polynomial differential systems, Nonlinear Dynam., 78 (2014), 1659-1681.  doi: 10.1007/s11071-014-1541-8.

[13]

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.

[14]

B. Qiu and H. Liang, Classification of global phase portrait of planar quintic quasi-homogeneous coprime polynomial systems, Qual. Theory Dyn. Syst., 16 (2017), 417-451.  doi: 10.1007/s12346-016-0199-7.

[15]

Y. TangL. Wang and X. Zhang, Center of planar quintic quasi-homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst., 35 (2015), 2177-2191.  doi: 10.3934/dcds.2015.35.2177.

[16]

Y. Tian and H. Liang, Planar semi-quasi homogeneous polynomial differential systems with a given degree, Qual. Theory Dyn. Syst., 18 (2019), 841-871.  doi: 10.1007/s12346-019-00316-w.

[17]

Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Volume 101 of Trans. of Mathematical Monographs, Am. Math. Soc. Providence, RI, 1992.

[18]

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.

Figure 1.  The global phase portraits of system (2) with $ a<-1 $
Figure 2.  The topological equivalence classes of the global phase portraits of planar cubic semi-quasi-homogeneous systems
Figure 6.  The global phase portraits of systems $ (A_{1}) $ and $ (B_{1, k})\ (k = 1, 2, 3) $
Figure 5.  Local phase portrait of system (10) at the origin
Figure 7.  The phase portrait of systems $ (M^{\pm}_{1, k, l}) $
Figure 8.  The global phase portraits of systems $ (A^{\pm}_{2, k}) $
Figure 9.  The global phase portraits of systems $ (C_{1, k}), $ $ (D_1), $ $ (E_{1, k}) $ and $ (F_{1}) $
Figure 10.  The global phase portraits of systems $ (G^{\pm}_{1, k, l}) $
Figure 11.  The global phase portraits of systems $ (H^{\pm}_{1, k}) $ and $ (I^{\pm}_{1, k}) $
Figure 12.  The phase portraits of systems $ (J^{\pm}_{1, k, l}) $ and $ L^{\pm}_{1, k, l}) $
Figure 13.  The global phase portraits of systems $ (B^{\pm}_{2, k}) $
Figure 14.  The global phase portraits of systems $ (C_{2, 1}) $ and $ (C^{\pm}_{2, 2}) $
Figure 15.  The global phase portraits of systems $ (D^{\pm}_{2, k}) $
Figure 16.  The global phase portraits of systems $ (E_{2, k}) $
Figure 17.  The global phase portraits of systems $ (F_{2, k}) $
Table 1.  The known results about the CH, CSH, CQH, and CSQH before this paper
Type of cubic systems The maximum number of limit cycles The global dynamics
Homogeneous 0 completed
Semi-Homogeneous ≥ 1 for some subclasses
Quasi-Homogeneous ? for some subclasses
Semi-Quasi-Homogeneous ? ?
Type of cubic systems The maximum number of limit cycles The global dynamics
Homogeneous 0 completed
Semi-Homogeneous ≥ 1 for some subclasses
Quasi-Homogeneous ? for some subclasses
Semi-Quasi-Homogeneous ? ?
Table 2.  The known results about the CH, CSH, CQH, and CSQH after this paper
Type of cubic systems The maximum number of limit cycles The global dynamics
Homogeneous 0 completed
Semi-Homogeneous 1 for some subclasses
Quasi-Homogeneous 0 for some subclasses
Semi-Quasi-Homogeneous 1 completed
Type of cubic systems The maximum number of limit cycles The global dynamics
Homogeneous 0 completed
Semi-Homogeneous 1 for some subclasses
Quasi-Homogeneous 0 for some subclasses
Semi-Quasi-Homogeneous 1 completed
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