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Chaos control in a special pendulum system for ultra-subharmonic resonance
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 |
$ \begin{align*} \frac{d}{dt}(x,y) = \boldsymbol X_n+\boldsymbol X_m, \end{align*} $ |
$ \boldsymbol X_n $ |
$ \boldsymbol X_m $ |
$ n $ |
$ m $ |
$ 1 $ |
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.
|

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