
-
Previous Article
Asymptotic behavior for stochastic plate equations with memory and additive noise on unbounded domains
- DCDS-B Home
- This Issue
-
Next Article
Persistence of mosquito vector and dengue: Impact of seasonal and diurnal temperature variations
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 |
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.
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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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.
|
[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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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.
|
[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. |
[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. 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. |
[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. |











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 | ? | ? |
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 |
[1] |
C. R. Zhu, K. Q. Lan. Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates. Discrete and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 2015, 35 (5) : 2177-2191. doi: 10.3934/dcds.2015.35.2177 |
[14] |
Andrew D. Barwell, Chris Good, Piotr Oprocha, Brian E. Raines. Characterizations of $\omega$-limit sets in topologically hyperbolic systems. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1819-1833. doi: 10.3934/dcds.2013.33.1819 |
[15] |
Maoan Han, Yuhai Wu, Ping Bi. A new cubic system having eleven limit cycles. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 675-686. doi: 10.3934/dcds.2005.12.675 |
[16] |
Jaume Giné, Maite Grau, Jaume Llibre. Polynomial and rational first integrals for planar quasi--homogeneous polynomial differential systems. Discrete and 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 and 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 and 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 and 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 and 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
[Back to Top]