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  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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

L. A. Čerkas, Number of limit cycles of an autonomous second-order system, Differ. Uravn., 12 (1976), 944-946.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. Llibre and G. Świrszcz, On the limit cycles of polynomial vector fields, Dyn. Contin. Discrete Impuls. Syst, 18 (2011), 203-214.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

K. S. Sibirskii, On the number of limit cycles in the neighborhood of a singular point, Differential Equations, 1 (1965), 53-66.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

L. A. Čerkas, Number of limit cycles of an autonomous second-order system, Differ. Uravn., 12 (1976), 944-946.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. Llibre and G. Świrszcz, On the limit cycles of polynomial vector fields, Dyn. Contin. Discrete Impuls. Syst, 18 (2011), 203-214.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

K. S. Sibirskii, On the number of limit cycles in the neighborhood of a singular point, Differential Equations, 1 (1965), 53-66.   Google Scholar

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 $
[1]

Yanqin Xiong, Maoan Han. Planar quasi-homogeneous polynomial systems with a given weight degree. Discrete & Continuous Dynamical Systems - A, 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 & 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 2010, 27 (1) : 217-229. doi: 10.3934/dcds.2010.27.217

[6]

Hebai Chen, Jaume Llibre, Yilei Tang. Centers of discontinuous piecewise smooth quasi–homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6495-6509. doi: 10.3934/dcdsb.2019150

[7]

Yilei Tang, Long Wang, Xiang Zhang. Center of planar quintic quasi--homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2177-2191. doi: 10.3934/dcds.2015.35.2177

[8]

Jaume Giné, Maite Grau, Jaume Llibre. Polynomial and rational first integrals for planar quasi--homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4531-4547. doi: 10.3934/dcds.2013.33.4531

[9]

Freddy Dumortier. Sharp upperbounds for the number of large amplitude limit cycles in polynomial Lienard systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1465-1479. doi: 10.3934/dcds.2012.32.1465

[10]

Jaume Llibre, Lucyjane de A. S. Menezes. On the limit cycles of a class of discontinuous piecewise linear differential systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1835-1858. doi: 10.3934/dcdsb.2020005

[11]

Salomón Rebollo-Perdomo, Claudio Vidal. Bifurcation of limit cycles for a family of perturbed Kukles differential systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4189-4202. doi: 10.3934/dcds.2018182

[12]

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

[13]

Jackson Itikawa, Jaume Llibre, Ana Cristina Mereu, Regilene Oliveira. Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3259-3272. doi: 10.3934/dcdsb.2017136

[14]

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 - A, 2017, 37 (6) : 3353-3386. doi: 10.3934/dcds.2017142

[15]

Min Li, Maoan Han. On the number of limit cycles of a quartic polynomial system. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020337

[16]

Jianhe Shen, Maoan Han. Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3085-3108. doi: 10.3934/dcds.2013.33.3085

[17]

Jaume Llibre, Ana Rodrigues. On the limit cycles of the Floquet differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1129-1136. doi: 10.3934/dcdsb.2014.19.1129

[18]

José Luis Bravo, Manuel Fernández, Ignacio Ojeda, Fernando Sánchez. Uniqueness of limit cycles for quadratic vector fields. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 483-502. doi: 10.3934/dcds.2019020

[19]

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

[20]

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

2019 Impact Factor: 1.27

Article outline

Figures and Tables

[Back to Top]