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

* Corresponding author: Jianfeng Huang

Received  January 2019 Revised  January 2020 Published  February 2021 Early access  May 2020

Fund Project: The first author is supported by the NSF of China (No.11401255, No.11771101) and the China Scholarship Council (No.201606785007). The second author is supported by the NSF of China (No.11771101), the major research program of colleges and universities in Guangdong Province (No.2017KZDXM054), and the Science and Technology Program of Guangzhou, China (201805010001)

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 $
and
$ \boldsymbol X_m $
are quasi-homogeneous vector fields of degree
$ n $
and
$ m $
respectively. We prove that under a new hypothesis, the maximal number of limit cycles of the system is
$ 1 $
. We also show that our result can be applied to some systems when the previous results are invalid. The proof is based on the investigations for the Abel equation and the generalized-polar equation associated with the system, respectively. Usually these two kinds of equations need to be dealt with separately, and for both equations, an efficient approach to estimate the number of periodic solutions is constructing suitable auxiliary functions. In the present paper we introduce a formula on the divergence, which allows us to construct an auxiliary function of one equation with the auxiliary function of the other equation, and vice versa.
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. AlgabaE. FreireE. 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. AlgabaE. 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. AlgabaC. 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. ÁlvarezA. 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. ChavarrigaI. 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. CimaA. 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. CollA. 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. GasullJ. 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. GavrilovJ. 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. GinéM. 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. HuangH. 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. LiJ. LlibreJ. 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. LlibreJesús 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. LlibreJesús 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. AlgabaE. FreireE. 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. AlgabaE. 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. AlgabaC. 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. ÁlvarezA. 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. ChavarrigaI. 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. CimaA. 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. CollA. 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. GasullJ. 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. GavrilovJ. 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. GinéM. 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. HuangH. 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. LiJ. LlibreJ. 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. LlibreJesús 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. LlibreJesús 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. 

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 $
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