Figure 1.
The dot curve represents the curve $ b_n(\theta)+b_m(\theta)r = 0 $ in Cartesian coordinates, which is the inner boundary of the region $ U $
-
Previous Article
On the rigid-lid approximation of shallow water Bingham
- DCDS-B Home
- This Issue
-
Next Article
Chaos control in a special pendulum system for ultra-subharmonic resonance
February
2021, 26(2): 861-873.
doi: 10.3934/dcdsb.2020145
Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields
| 1. | Department of Mathematics, Jinan University, Guangzhou 510632, P.R. China |
| 2. | School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou 510665, P.R. China |
In this paper we consider the limit cycles of the planar system
| $ \begin{align*} \frac{d}{dt}(x,y) = \boldsymbol X_n+\boldsymbol X_m, \end{align*} $ |
where
| $ \boldsymbol X_n $ |
| $ \boldsymbol X_m $ |
| $ n $ |
| $ m $ |
| $ 1 $ |
Keywords:
Limit cycles,
existence and uniqueness,
polynomial differential systems,
quasi-homogeneous nonlinearities.
Mathematics Subject Classification:
Primary: 34C07; Secondary: 37C05; Tertiary: 35F60.
Citation:
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
![]()
References:
| [1] |
A. Algaba, E. Freire, E. Gamero and C. García,
Monodromy, center-focus and integrability problems for quasi-homogeneous polynomial systems, Nonlinear Anal., 72 (2010), 1726-1736.
doi: 10.1016/j.na.2009.09.012. |
| [2] |
A. Algaba, E. Gamero and C. García,
The integrability problem for a class of planar systems, Nonlinearity, 22 (2009), 395-420.
doi: 10.1088/0951-7715/22/2/009. |
| [3] |
A. Algaba, C. García and M. Reyes,
Integrability of two dimensional quasi-homogeneous polynomial differential systems, Rocky Mt. J. Math., 41 (2011), 1-22.
doi: 10.1216/RMJ-2011-41-1-1. |
| [4] |
M. J. Álvarez, A. Gasull and H. Giacomini,
A new uniqueness criterion for the number of periodic orbits of Abel equations, J. Differential Equations, 234 (2007), 161-176.
doi: 10.1016/j.jde.2006.11.004. |
| [5] |
R. Benterki and J. Llibre,
Limit cycles of polynomial differential equations with quintic homogeneous nonlinearities, J. Math. Anal. Appl., 407 (2013), 16-22.
doi: 10.1016/j.jmaa.2013.04.076. |
| [6] |
L. Cairó and J. Llibre,
Phase portraits of planar semi-homogeneous vector fields (Ⅰ), Nonlinear Anal., 29 (1997), 783-811.
doi: 10.1016/S0362-546X(96)00088-0. |
| [7] |
L. Cairó and J. Llibre,
Phase portraits of planar semi-homogeneous vector fields (Ⅱ), Nonlinear Anal., 39 (2000), 351-363.
doi: 10.1016/S0362-546X(98)00177-1. |
| [8] |
M. Carbonell and J. Llibre,
Limit cycles of a class of polynomial systems, Proc. Royal Soc. Edinburgh, 109 (1988), 187-199.
doi: 10.1017/S0308210500026755. |
| [9] |
J. Chavarriga, I. A. Garcia and J. Gine,
On integrability of differential equations defined by the sum of homogeneous vector fields with degenerate infinity, Int. J. Bifurcation Chaos, 11 (2011), 711-722.
doi: 10.1142/S0218127401002390. |
| [10] |
L. A. Čerkas,
Number of limit cycles of an autonomous second-order system, Differ. Uravn., 12 (1976), 944-946.
|
| [11] |
A. Cima, A. Gasull and F. Mańosas,
Limit cycles for vector fields with homogeneous components, Appl. Math., 24 (1997), 281-287.
doi: 10.4064/am-24-3-281-287. |
| [12] |
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. |
| [13] |
B. Coll, A. Gasull and R. Prohens,
Differential equations defined by the sum of two quasi-homogeneous vector fields, Can. J. Math., 49 (1997), 212-231.
doi: 10.4153/CJM-1997-011-0. |
| [14] |
A. Gasull and J. Llibre,
Limit cycles for a class of Abel equation, SIAM J. Math. Anal., 21 (1990), 1235-1244.
doi: 10.1137/0521068. |
| [15] |
A. Gasull, J. Yu and X. Zhang,
Vector fields with homogeneous nonlinearities and many limit cycles, J. Differential Equations, 258 (2015), 3286-3303.
doi: 10.1016/j.jde.2015.01.009. |
| [16] |
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. |
| [17] |
J. Giné, M. Grau and J. Llibre,
Limit cycles bifurcating from planar polynomial quasi-homogeneous centers, J. Differential Equations, 259 (2015), 7135-7160.
doi: 10.1016/j.jde.2015.08.014. |
| [18] |
J. Huang and H. Liang,
A uniqueness criterion of limit cycles for planar polynomial systems with homogeneous nonlinearities, J. Math. Anal. Appl., 457 (2018), 498-521.
doi: 10.1016/j.jmaa.2017.08.008. |
| [19] |
J. Huang, H. Liang and J. Llibre,
Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities, J. Differential Equations, 265 (2018), 3888-3913.
doi: 10.1016/j.jde.2018.05.019. |
| [20] |
J. Huang and Y. Zhao,
Periodic solutions for equation $\dot{x}=A(t)x^m+B(t)x^n+C(t)x^l$ with $A(t)$ and $B(t)$ changing signs, J. Differential Equations, 253 (2012), 73-99.
doi: 10.1016/j.jde.2012.03.021. |
| [21] |
W. Li, J. Llibre, J. Yang and Z. Zhang,
Limit cycles bifurcating from the period annulus of quasi-homogeneous centers, J. Dynam. Differential Equations, 21 (2009), 133-152.
doi: 10.1007/s10884-008-9126-1. |
| [22] |
J. Llibre, Jesús S. Pérez del Río and J. A. Rodríguez,
Structural stability of planar homogeneous polynomial vector fields: Applications to critical points and to infinity, J. Differential Equations, 125 (1996), 490-520.
doi: 10.1006/jdeq.1996.0038. |
| [23] |
J. Llibre, Jesús S. Pérez del Río and J. A. Rodríguez,
Structural stability of planar semi-homogeneous polynomial vector fields: Applications to critical points and to infinity, Discrete Contin. Dyn. Syst., 6 (2000), 809-828.
doi: 10.3934/dcds.2000.6.809. |
| [24] |
J. Llibre and G. Świrszcz,
On the limit cycles of polynomial vector fields, Dyn. Contin. Discrete Impuls. Syst, 18 (2011), 203-214.
|
| [25] |
J. Llibre and C. Valls,
Classification of the centers, their cyclicity and isochronicity for a class of polynomial differential systems generalizing the linear systems with cubic homogeneous nonlinearities, J. Differential Equations, 246 (2009), 2192-2204.
doi: 10.1016/j.jde.2008.12.006. |
| [26] |
N. G. Lloyd,
A note on the number of limit cycles in certain two-dimensional systems, J. London Math. Soc., 20 (1979), 277-286.
doi: 10.1112/jlms/s2-20.2.277. |
| [27] |
N. G. Lloyd and J. M. Pearson,
Bifurcation of limit cycles and integrability of planar dynamical systems in complex form, J. Phys. A: Math. Gen., 32 (1999), 1973-1984.
doi: 10.1088/0305-4470/32/10/014. |
| [28] |
K. S. Sibirskii,
On the number of limit cycles in the neighborhood of a singular point, Differential Equations, 1 (1965), 53-66.
|
show all references
References:
| [1] |
A. Algaba, E. Freire, E. Gamero and C. García,
Monodromy, center-focus and integrability problems for quasi-homogeneous polynomial systems, Nonlinear Anal., 72 (2010), 1726-1736.
doi: 10.1016/j.na.2009.09.012. |
| [2] |
A. Algaba, E. Gamero and C. García,
The integrability problem for a class of planar systems, Nonlinearity, 22 (2009), 395-420.
doi: 10.1088/0951-7715/22/2/009. |
| [3] |
A. Algaba, C. García and M. Reyes,
Integrability of two dimensional quasi-homogeneous polynomial differential systems, Rocky Mt. J. Math., 41 (2011), 1-22.
doi: 10.1216/RMJ-2011-41-1-1. |
| [4] |
M. J. Álvarez, A. Gasull and H. Giacomini,
A new uniqueness criterion for the number of periodic orbits of Abel equations, J. Differential Equations, 234 (2007), 161-176.
doi: 10.1016/j.jde.2006.11.004. |
| [5] |
R. Benterki and J. Llibre,
Limit cycles of polynomial differential equations with quintic homogeneous nonlinearities, J. Math. Anal. Appl., 407 (2013), 16-22.
doi: 10.1016/j.jmaa.2013.04.076. |
| [6] |
L. Cairó and J. Llibre,
Phase portraits of planar semi-homogeneous vector fields (Ⅰ), Nonlinear Anal., 29 (1997), 783-811.
doi: 10.1016/S0362-546X(96)00088-0. |
| [7] |
L. Cairó and J. Llibre,
Phase portraits of planar semi-homogeneous vector fields (Ⅱ), Nonlinear Anal., 39 (2000), 351-363.
doi: 10.1016/S0362-546X(98)00177-1. |
| [8] |
M. Carbonell and J. Llibre,
Limit cycles of a class of polynomial systems, Proc. Royal Soc. Edinburgh, 109 (1988), 187-199.
doi: 10.1017/S0308210500026755. |
| [9] |
J. Chavarriga, I. A. Garcia and J. Gine,
On integrability of differential equations defined by the sum of homogeneous vector fields with degenerate infinity, Int. J. Bifurcation Chaos, 11 (2011), 711-722.
doi: 10.1142/S0218127401002390. |
| [10] |
L. A. Čerkas,
Number of limit cycles of an autonomous second-order system, Differ. Uravn., 12 (1976), 944-946.
|
| [11] |
A. Cima, A. Gasull and F. Mańosas,
Limit cycles for vector fields with homogeneous components, Appl. Math., 24 (1997), 281-287.
doi: 10.4064/am-24-3-281-287. |
| [12] |
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. |
| [13] |
B. Coll, A. Gasull and R. Prohens,
Differential equations defined by the sum of two quasi-homogeneous vector fields, Can. J. Math., 49 (1997), 212-231.
doi: 10.4153/CJM-1997-011-0. |
| [14] |
A. Gasull and J. Llibre,
Limit cycles for a class of Abel equation, SIAM J. Math. Anal., 21 (1990), 1235-1244.
doi: 10.1137/0521068. |
| [15] |
A. Gasull, J. Yu and X. Zhang,
Vector fields with homogeneous nonlinearities and many limit cycles, J. Differential Equations, 258 (2015), 3286-3303.
doi: 10.1016/j.jde.2015.01.009. |
| [16] |
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. |
| [17] |
J. Giné, M. Grau and J. Llibre,
Limit cycles bifurcating from planar polynomial quasi-homogeneous centers, J. Differential Equations, 259 (2015), 7135-7160.
doi: 10.1016/j.jde.2015.08.014. |
| [18] |
J. Huang and H. Liang,
A uniqueness criterion of limit cycles for planar polynomial systems with homogeneous nonlinearities, J. Math. Anal. Appl., 457 (2018), 498-521.
doi: 10.1016/j.jmaa.2017.08.008. |
| [19] |
J. Huang, H. Liang and J. Llibre,
Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities, J. Differential Equations, 265 (2018), 3888-3913.
doi: 10.1016/j.jde.2018.05.019. |
| [20] |
J. Huang and Y. Zhao,
Periodic solutions for equation $\dot{x}=A(t)x^m+B(t)x^n+C(t)x^l$ with $A(t)$ and $B(t)$ changing signs, J. Differential Equations, 253 (2012), 73-99.
doi: 10.1016/j.jde.2012.03.021. |
| [21] |
W. Li, J. Llibre, J. Yang and Z. Zhang,
Limit cycles bifurcating from the period annulus of quasi-homogeneous centers, J. Dynam. Differential Equations, 21 (2009), 133-152.
doi: 10.1007/s10884-008-9126-1. |
| [22] |
J. Llibre, Jesús S. Pérez del Río and J. A. Rodríguez,
Structural stability of planar homogeneous polynomial vector fields: Applications to critical points and to infinity, J. Differential Equations, 125 (1996), 490-520.
doi: 10.1006/jdeq.1996.0038. |
| [23] |
J. Llibre, Jesús S. Pérez del Río and J. A. Rodríguez,
Structural stability of planar semi-homogeneous polynomial vector fields: Applications to critical points and to infinity, Discrete Contin. Dyn. Syst., 6 (2000), 809-828.
doi: 10.3934/dcds.2000.6.809. |
| [24] |
J. Llibre and G. Świrszcz,
On the limit cycles of polynomial vector fields, Dyn. Contin. Discrete Impuls. Syst, 18 (2011), 203-214.
|
| [25] |
J. Llibre and C. Valls,
Classification of the centers, their cyclicity and isochronicity for a class of polynomial differential systems generalizing the linear systems with cubic homogeneous nonlinearities, J. Differential Equations, 246 (2009), 2192-2204.
doi: 10.1016/j.jde.2008.12.006. |
| [26] |
N. G. Lloyd,
A note on the number of limit cycles in certain two-dimensional systems, J. London Math. Soc., 20 (1979), 277-286.
doi: 10.1112/jlms/s2-20.2.277. |
| [27] |
N. G. Lloyd and J. M. Pearson,
Bifurcation of limit cycles and integrability of planar dynamical systems in complex form, J. Phys. A: Math. Gen., 32 (1999), 1973-1984.
doi: 10.1088/0305-4470/32/10/014. |
| [28] |
K. S. Sibirskii,
On the number of limit cycles in the neighborhood of a singular point, Differential Equations, 1 (1965), 53-66.
|
| [1] |
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 |
| [2] |
Jaume Llibre, Claudia Valls. Algebraic limit cycles for quadratic polynomial differential systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2475-2485. doi: 10.3934/dcdsb.2018070 |
| [3] |
Antonio Algaba, Estanislao Gamero, Cristóbal García. The reversibility problem for quasi-homogeneous dynamical systems. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3225-3236. doi: 10.3934/dcds.2013.33.3225 |
| [4] |
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 |
| [5] |
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 |
| [6] |
Tao Li, Jaume Llibre. Limit cycles of piecewise polynomial differential systems with the discontinuity line xy = 0. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3887-3909. doi: 10.3934/cpaa.2021136 |
| [7] |
Hebai Chen, Jaume Llibre, Yilei Tang. Centers of discontinuous piecewise smooth quasi–homogeneous polynomial differential systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6495-6509. doi: 10.3934/dcdsb.2019150 |
| [8] |
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 |
| [9] |
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 |
| [10] |
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 |
| [11] |
Freddy Dumortier. Sharp upperbounds for the number of large amplitude limit cycles in polynomial Lienard systems. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1465-1479. doi: 10.3934/dcds.2012.32.1465 |
| [12] |
Jaume Llibre, Lucyjane de A. S. Menezes. On the limit cycles of a class of discontinuous piecewise linear differential systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1835-1858. doi: 10.3934/dcdsb.2020005 |
| [13] |
Salomón Rebollo-Perdomo, Claudio Vidal. Bifurcation of limit cycles for a family of perturbed Kukles differential systems. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4189-4202. doi: 10.3934/dcds.2018182 |
| [14] |
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 |
| [15] |
Jackson Itikawa, Jaume Llibre, Ana Cristina Mereu, Regilene Oliveira. Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3259-3272. doi: 10.3934/dcdsb.2017136 |
| [16] |
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 |
| [17] |
Min Li, Maoan Han. On the number of limit cycles of a quartic polynomial system. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3167-3181. doi: 10.3934/dcdss.2020337 |
| [18] |
Jianhe Shen, Maoan Han. Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3085-3108. doi: 10.3934/dcds.2013.33.3085 |
| [19] |
Jaume Llibre, Ana Rodrigues. On the limit cycles of the Floquet differential equation. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1129-1136. doi: 10.3934/dcdsb.2014.19.1129 |
| [20] |
José Luis Bravo, Manuel Fernández, Ignacio Ojeda, Fernando Sánchez. Uniqueness of limit cycles for quadratic vector fields. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 483-502. doi: 10.3934/dcds.2019020 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]