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

Previous Article
Stable transition layer induced by degeneracy of the spatial inhomogeneities in the Allen-Cahn problem
DCDS-B Home
This Issue
Next Article
Turing instability and pattern formations for reaction-diffusion systems on 2D bounded domain

doi: 10.3934/dcdsb.2021049

Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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

Zecen He ^1, , Haihua Liang ^1,, and Xiang Zhang ^2,

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 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.

Keywords: Semi-quasi-homogeneous, planar cubic systems, limit cycles, topologically equivalent, global phase portraits.

Mathematics Subject Classification: Primary: 34C07; Secondary: 34C05, 34C41.

Citation: Zecen He, Haihua Liang, Xiang Zhang. Limit cycles and global dynamic of planar cubic semi-quasi-homogeneous systems. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021049

References:

[1]	A. Algaba, C. 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. Google Scholar
[2]	W. Aziz, J. 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. Google Scholar
[3]	L. Cairó and J. Llibre, Phase portraits of planar semi-homogeneous vector fields (Ⅱ), Nonlinear Anal., 39 (2000), 351-363. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	A. Cima, A. 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. Google Scholar
[7]	A. Cima, A. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[10]	B. García, J. 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. Google Scholar
[11]	G. Huang, G. 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. Google Scholar
[12]	H. Liang, J. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	Y. Tang, L. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

show all references

References:

[1]	A. Algaba, C. 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. Google Scholar
[2]	W. Aziz, J. 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. Google Scholar
[3]	L. Cairó and J. Llibre, Phase portraits of planar semi-homogeneous vector fields (Ⅱ), Nonlinear Anal., 39 (2000), 351-363. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	A. Cima, A. 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. Google Scholar
[7]	A. Cima, A. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[10]	B. García, J. 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. Google Scholar
[11]	G. Huang, G. 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. Google Scholar
[12]	H. Liang, J. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	Y. Tang, L. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

Figure 1. The global phase portraits of system (2) with $ a<-1 $

Download full-size image

Download as PowerPoint slide

Figure 2. The topological equivalence classes of the global phase portraits of planar cubic semi-quasi-homogeneous systems

Download full-size image

Download as PowerPoint slide

Figure 3.

Download full-size image

Download as PowerPoint slide

Figure 4.

Download full-size image

Download as PowerPoint slide

Figure 6. The global phase portraits of systems $ (A_{1}) $ and $ (B_{1, k})\ (k = 1, 2, 3) $

Download full-size image

Download as PowerPoint slide

Figure 5. Local phase portrait of system (10) at the origin

Download full-size image

Download as PowerPoint slide

Figure 7. The phase portrait of systems $ (M^{\pm}_{1, k, l}) $

Download full-size image

Download as PowerPoint slide

Figure 8. The global phase portraits of systems $ (A^{\pm}_{2, k}) $

Download full-size image

Download as PowerPoint slide

Figure 9. The global phase portraits of systems $ (C_{1, k}), $

$ (D_1), $

$ (E_{1, k}) $ and $ (F_{1}) $

Download full-size image

Download as PowerPoint slide

Figure 10. The global phase portraits of systems $ (G^{\pm}_{1, k, l}) $

Download full-size image

Download as PowerPoint slide

Figure 11. The global phase portraits of systems $ (H^{\pm}_{1, k}) $ and $ (I^{\pm}_{1, k}) $

Download full-size image

Download as PowerPoint slide

Figure 12. The phase portraits of systems $ (J^{\pm}_{1, k, l}) $ and $ L^{\pm}_{1, k, l}) $

Download full-size image

Download as PowerPoint slide

Figure 13. The global phase portraits of systems $ (B^{\pm}_{2, k}) $

Download full-size image

Download as PowerPoint slide

Figure 14. The global phase portraits of systems $ (C_{2, 1}) $ and $ (C^{\pm}_{2, 2}) $

Download full-size image

Download as PowerPoint slide

Figure 15. The global phase portraits of systems $ (D^{\pm}_{2, k}) $

Download full-size image

Download as PowerPoint slide

Figure 16. The global phase portraits of systems $ (E_{2, k}) $

Download full-size image

Download as PowerPoint slide

Figure 17. The global phase portraits of systems $ (F_{2, k}) $

Download full-size image

Download as PowerPoint slide

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	?	?

Download as excel

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

Download as excel

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

[1]	C. R. Zhu, K. Q. Lan. Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 289-306. doi: 10.3934/dcdsb.2010.14.289
[2]	Jianfeng Huang, Haihua Liang. Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 861-873. doi: 10.3934/dcdsb.2020145
[3]	Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214
[4]	Jackson Itikawa, Jaume Llibre. Global phase portraits of uniform isochronous centers with quartic homogeneous polynomial nonlinearities. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 121-131. doi: 10.3934/dcdsb.2016.21.121
[5]	Yilei Tang. Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2029-2046. doi: 10.3934/dcds.2018082
[6]	Victoriano Carmona, Soledad Fernández-García, Antonio E. Teruel. Saddle-node of limit cycles in planar piecewise linear systems and applications. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5275-5299. doi: 10.3934/dcds.2019215
[7]	Song-Mei Huan, Xiao-Song Yang. On the number of limit cycles in general planar piecewise linear systems. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2147-2164. doi: 10.3934/dcds.2012.32.2147
[8]	Armengol Gasull, Hector Giacomini. Upper bounds for the number of limit cycles of some planar polynomial differential systems. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 217-229. doi: 10.3934/dcds.2010.27.217
[9]	Shimin Li, Jaume Llibre. On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5885-5901. doi: 10.3934/dcdsb.2019111
[10]	Ricardo M. Martins, Otávio M. L. Gomide. Limit cycles for quadratic and cubic planar differential equations under polynomial perturbations of small degree. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3353-3386. doi: 10.3934/dcds.2017142
[11]	P. Yu, M. Han. Twelve limit cycles in a cubic order planar system with $Z_2$- symmetry. Communications on Pure & Applied Analysis, 2004, 3 (3) : 515-526. doi: 10.3934/cpaa.2004.3.515
[12]	Yanqin Xiong, Maoan Han. Planar quasi-homogeneous polynomial systems with a given weight degree. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 4015-4025. doi: 10.3934/dcds.2016.36.4015
[13]	Yilei Tang, Long Wang, Xiang Zhang. Center of planar quintic quasi--homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2177-2191. doi: 10.3934/dcds.2015.35.2177
[14]	Maoan Han, Yuhai Wu, Ping Bi. A new cubic system having eleven limit cycles. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 675-686. doi: 10.3934/dcds.2005.12.675
[15]	Andrew D. Barwell, Chris Good, Piotr Oprocha, Brian E. Raines. Characterizations of $\omega$-limit sets in topologically hyperbolic systems. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1819-1833. doi: 10.3934/dcds.2013.33.1819
[16]	Jaume Giné, Maite Grau, Jaume Llibre. Polynomial and rational first integrals for planar quasi--homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4531-4547. doi: 10.3934/dcds.2013.33.4531
[17]	Hebai Chen, Xingwu Chen. Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ). Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4141-4170. doi: 10.3934/dcdsb.2018130
[18]	K. Q. Lan, C. R. Zhu. Phase portraits of predator--prey systems with harvesting rates. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 901-933. doi: 10.3934/dcds.2012.32.901
[19]	Jaume Llibre, Yilei Tang. Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1769-1784. doi: 10.3934/dcdsb.2018236
[20]	Junmin Yang, Maoan Han. On the number of limit cycles of a cubic Near-Hamiltonian system. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 827-840. doi: 10.3934/dcds.2009.24.827

2020 Impact Factor: 1.327

Tools

Metrics

Other articles
by authors

on AIMS
Zecen He Haihua Liang Xiang Zhang
on Google Scholar
Zecen He Haihua Liang Xiang Zhang

2020 Impact Factor: 1.327

Tools

Article outline

Figures and Tables

2020 Impact Factor: 1.327

Tools

Figures and Tables

2020 Impact Factor: 1.327

Tools

Metrics

Other articles
by authors

on AIMS
Zecen He Haihua Liang Xiang Zhang
on Google Scholar
Zecen He Haihua Liang Xiang Zhang

[Back to Top]