doi: 10.3934/dcdsb.2021314
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

The study on cyclicity of a class of cubic systems

1. 

School of Mathematical Sciences, CMA-Shanghai, Shanghai Jiaotong University, Shanghai 200240, China

2. 

College of Science, Zhongyuan University of Technology, Zhengzhou 450000, China

* Corresponding author: Jiang Yu

Received  July 2021 Revised  November 2021 Early access January 2022

Fund Project: The corresponding author is supported by the National Natural Science Foundations of China (11771282 and 11931016), Science and Technology Innovation Action Program of STCSM(20JC1413200), Innovation Program of Shanghai Municipal Education Commission (No. 2021-01-07-00-02-E00087). The first author is supported by the National Natural Science Foundations of China(11771282 and 12071285)

In this paper, we consider a class of cubic systems with polynomial perturbation of the degree at most $ n $, and estimate the upper bound of the number of isolated zeros of its Abelian integral. Furthermore, we obtain the distributions of limit cycles bifurcated from a $ Z_4 $-equivariant system with $ 5 $ centers.

Citation: Yuanyuan Chen, Jiang Yu. The study on cyclicity of a class of cubic systems. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021314
References:
[1]

H. ChenS. DuanY. Tang and J. Xie, Global dynamics of a mechanical system with dry friction, J. Differential Equations, 265 (2018), 5490-5519.  doi: 10.1016/j.jde.2018.06.013.

[2]

H. Chen and J. XIE, The number of limit cycles of the FitzHugh nerve system, Quart. Appl. Math., 73 (2015), 365-378.  doi: 10.1090/S0033-569X-2015-01384-7.

[3]

H. Chen and L. Zou, Global study of Rayleigh-Duffing oscillators, J. Phys. A, 49 (2016), 165202, 35 pp. doi: 10.1088/1751-8113/49/16/165202.

[4]

Y. Chen and C. Song, Stability and Hopf bifurcation analysis in a prey-predator system with stage-structure for prey and time delay, Chaos, Solitons Fractals, 38 (2008), 1104-1114.  doi: 10.1016/j.chaos.2007.01.035.

[5]

Y. ChenJ. Yu and C. Sun, Stability and Hopf bifurcation analysis in a three-level food chain system with delay, Chaos Solitons Fractals, 31 (2007), 683-694.  doi: 10.1016/j.chaos.2005.10.020.

[6]

C. Christopher and C. Li, Limit cycles of differential equations, CRM Barcelona. Birkhäuser Verlag, Basel, 2007.

[7]

C. B. Collins, Static stars: Some mathematical curiosities, Journal of Mathematical Physics, 18 (1977), 1374-1377.  doi: 10.1063/1.523431.

[8]

L. Gavrilov and I. D. Iliev, Quadratic perturbations of quadratic codimension-four centers, J. Math. Anal. Appl., 357 (2009), 69-76.  doi: 10.1016/j.jmaa.2009.04.004.

[9]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.

[10]

M. Han, Bifurcation Theory of Limit Cycles, Science Press Beijing, Beijing; Oxford, 2017.

[11]

X. Jiang, Z. She, Z. Feng and X. Zheng, Bifurcation Analysis of a Predator-Prey System with Ratio-Dependent Functional Response, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750222, 21 pp. doi: 10.1142/S0218127417502224.

[12]

C. LiC. Liu and J. Yang, A cubic system with thirteen limit cycles, J. Differential Equations, 246 (2009), 3609-3619.  doi: 10.1016/j.jde.2009.01.038.

[13]

J. Li and Q. Huang, Bifurcations of limit cycles forming compound eyes in the cubic system, Chinese Ann. Math. Ser. B, 8 (1987), 391-403. 

[14]

J. Li and Y. Liu, New results on the study of Zq-equivariant planar polynomial vector fields, Qual. Theory Dyn. Syst., 9 (2010), 167-219.  doi: 10.1007/s12346-010-0024-7.

[15]

T. Puu, Attractors, Bifurcations & Chaos Nonlinear Phenomena in Economics, , Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-662-04094-2.

[16]

S. Wang, X. Wang and X. Wu, Bifurcation analysis for a food chain model with nonmonotonic nutrition conversion rate of predator to top predator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050113, 19 pp. doi: 10.1142/S0218127420501138.

[17]

J. Yang, On the number of zeros of Abelian integral for a class of cubic Hamiltonian systems, Dyn. Syst., 34 (2019), 561-583.  doi: 10.1080/14689367.2019.1574716.

[18]

J. YangS. Sui and L. Zhao, On the number of zeros of Abelian integral for a class of cubic Hamilton systems with the phase portrait "butterfly", Qual. Theory Dyn. Syst., 18 (2019), 947-967.  doi: 10.1007/s12346-019-00321-z.

[19]

J. Yang and L. Zhao, The cyclicity of period annuli for a class of cubic Hamiltonian systems with nilpotent singular points, J. Differential Equations, 263 (2017), 5554-5581.  doi: 10.1016/j.jde.2017.06.027.

[20]

J. YangM. HanJ. Li and P. Yu, Existence conditions of thirteen limit cycles in a cubic system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2569-2577.  doi: 10.1142/S0218127410027209.

[21]

P. Yu and M. Han, Twelve limit cycles in a cubic case of the 16th Hilbert problem, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 2191-2205.  doi: 10.1142/S0218127405013289.

[22]

Y. Zhao and Z. Zhang, Linear estimate of the number of zeros of abelian integrals for a kind of quartic Hamiltonians, J. Differential Equations, 155 (1999), 73-88.  doi: 10.1006/jdeq.1998.3581.

[23]

X. Zhou and C. Li, Estimate of the number of zeros of Abelian integrals for a kind of quartic Hamiltonians with two centers, Appl. Math. Comput., 204 (2008), 202-209.  doi: 10.1016/j.amc.2008.06.036.

[24]

X. Zhou and C. Li, On the algebraic structure of Abelian integrals for a kind of perturbed cubic Hamiltonian systems, J. Math. Anal. Appl., 359 (2009), 209-215.  doi: 10.1016/j.jmaa.2009.05.034.

show all references

References:
[1]

H. ChenS. DuanY. Tang and J. Xie, Global dynamics of a mechanical system with dry friction, J. Differential Equations, 265 (2018), 5490-5519.  doi: 10.1016/j.jde.2018.06.013.

[2]

H. Chen and J. XIE, The number of limit cycles of the FitzHugh nerve system, Quart. Appl. Math., 73 (2015), 365-378.  doi: 10.1090/S0033-569X-2015-01384-7.

[3]

H. Chen and L. Zou, Global study of Rayleigh-Duffing oscillators, J. Phys. A, 49 (2016), 165202, 35 pp. doi: 10.1088/1751-8113/49/16/165202.

[4]

Y. Chen and C. Song, Stability and Hopf bifurcation analysis in a prey-predator system with stage-structure for prey and time delay, Chaos, Solitons Fractals, 38 (2008), 1104-1114.  doi: 10.1016/j.chaos.2007.01.035.

[5]

Y. ChenJ. Yu and C. Sun, Stability and Hopf bifurcation analysis in a three-level food chain system with delay, Chaos Solitons Fractals, 31 (2007), 683-694.  doi: 10.1016/j.chaos.2005.10.020.

[6]

C. Christopher and C. Li, Limit cycles of differential equations, CRM Barcelona. Birkhäuser Verlag, Basel, 2007.

[7]

C. B. Collins, Static stars: Some mathematical curiosities, Journal of Mathematical Physics, 18 (1977), 1374-1377.  doi: 10.1063/1.523431.

[8]

L. Gavrilov and I. D. Iliev, Quadratic perturbations of quadratic codimension-four centers, J. Math. Anal. Appl., 357 (2009), 69-76.  doi: 10.1016/j.jmaa.2009.04.004.

[9]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.

[10]

M. Han, Bifurcation Theory of Limit Cycles, Science Press Beijing, Beijing; Oxford, 2017.

[11]

X. Jiang, Z. She, Z. Feng and X. Zheng, Bifurcation Analysis of a Predator-Prey System with Ratio-Dependent Functional Response, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750222, 21 pp. doi: 10.1142/S0218127417502224.

[12]

C. LiC. Liu and J. Yang, A cubic system with thirteen limit cycles, J. Differential Equations, 246 (2009), 3609-3619.  doi: 10.1016/j.jde.2009.01.038.

[13]

J. Li and Q. Huang, Bifurcations of limit cycles forming compound eyes in the cubic system, Chinese Ann. Math. Ser. B, 8 (1987), 391-403. 

[14]

J. Li and Y. Liu, New results on the study of Zq-equivariant planar polynomial vector fields, Qual. Theory Dyn. Syst., 9 (2010), 167-219.  doi: 10.1007/s12346-010-0024-7.

[15]

T. Puu, Attractors, Bifurcations & Chaos Nonlinear Phenomena in Economics, , Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-662-04094-2.

[16]

S. Wang, X. Wang and X. Wu, Bifurcation analysis for a food chain model with nonmonotonic nutrition conversion rate of predator to top predator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050113, 19 pp. doi: 10.1142/S0218127420501138.

[17]

J. Yang, On the number of zeros of Abelian integral for a class of cubic Hamiltonian systems, Dyn. Syst., 34 (2019), 561-583.  doi: 10.1080/14689367.2019.1574716.

[18]

J. YangS. Sui and L. Zhao, On the number of zeros of Abelian integral for a class of cubic Hamilton systems with the phase portrait "butterfly", Qual. Theory Dyn. Syst., 18 (2019), 947-967.  doi: 10.1007/s12346-019-00321-z.

[19]

J. Yang and L. Zhao, The cyclicity of period annuli for a class of cubic Hamiltonian systems with nilpotent singular points, J. Differential Equations, 263 (2017), 5554-5581.  doi: 10.1016/j.jde.2017.06.027.

[20]

J. YangM. HanJ. Li and P. Yu, Existence conditions of thirteen limit cycles in a cubic system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2569-2577.  doi: 10.1142/S0218127410027209.

[21]

P. Yu and M. Han, Twelve limit cycles in a cubic case of the 16th Hilbert problem, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 2191-2205.  doi: 10.1142/S0218127405013289.

[22]

Y. Zhao and Z. Zhang, Linear estimate of the number of zeros of abelian integrals for a kind of quartic Hamiltonians, J. Differential Equations, 155 (1999), 73-88.  doi: 10.1006/jdeq.1998.3581.

[23]

X. Zhou and C. Li, Estimate of the number of zeros of Abelian integrals for a kind of quartic Hamiltonians with two centers, Appl. Math. Comput., 204 (2008), 202-209.  doi: 10.1016/j.amc.2008.06.036.

[24]

X. Zhou and C. Li, On the algebraic structure of Abelian integrals for a kind of perturbed cubic Hamiltonian systems, J. Math. Anal. Appl., 359 (2009), 209-215.  doi: 10.1016/j.jmaa.2009.05.034.

Figure 1.  Phase portraits of system (1.8) for $ c>0 $. $ D_{1}^{+} = \{a>0, b^2 < 4ac\}, D_{2}^{+} = \{a<0, b>2a\}, D_{3}^{+} = \{a<0, b<2a\}, D_{4}^{+} = \{a<0, b^2>4ac\}, l_{1}^{+} = \{b = 2a, a<0\}, l_{2}^{+} = \{b^2 = 4ac, a>0\} $
Figure 2.  Phase portraits of system (1.8) for $ c<0 $. $ D_{1}^{-} = \{a>0,b\geq 2c\}, D_{2}^{-} = \{a<0,b^2>4ac\}, D_{3}^{-} = \{a<0,b^2<4ac,b>2c,b>2a\}, D_{4}^{-} = \{a<0,b\leq 2c,b\geq 2a\}, D_{5}^{-} = \{a<0,b\geq 2c,b\leq2a\},D_{6}^{-} = \{a<0,b<2c,b<2a\}, D_{7}^{-} = \{a>0,b<2c\}, l_{1}^{-} = \{b = 2a, a<0\}, l_{2}^{-} = \{b^2 = 4ac,a<0\} $
Figure 3.  The distribution of 5 limit cycles of system (1.10)
Figure 4.  $ D_{3(2)}^{-} $
Figure 5.  Phase portrait of unperturbed system of (1.10)
Figure 6.  The distribution of limit cycles when $ r(h) = \alpha $
[1]

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

[2]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

[3]

Jaume Llibre, Claudia Valls. Rational limit cycles of Abel equations. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1077-1089. doi: 10.3934/cpaa.2021007

[4]

Yunming Zhou, Desheng Shang, Tonghua Zhang. Seventeen limit cycles bifurcations of a fifth system. Conference Publications, 2007, 2007 (Special) : 1070-1081. doi: 10.3934/proc.2007.2007.1070

[5]

Jaume Llibre, Dana Schlomiuk. On the limit cycles bifurcating from an ellipse of a quadratic center. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1091-1102. doi: 10.3934/dcds.2015.35.1091

[6]

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

[7]

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

[8]

Maoan Han, Tonghua Zhang. Some bifurcation methods of finding limit cycles. Mathematical Biosciences & Engineering, 2006, 3 (1) : 67-77. doi: 10.3934/mbe.2006.3.67

[9]

Zhanyuan Hou, Stephen Baigent. Heteroclinic limit cycles in competitive Kolmogorov systems. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4071-4093. doi: 10.3934/dcds.2013.33.4071

[10]

Maoan Han. On some properties and limit cycles of Lienard systems. Conference Publications, 2001, 2001 (Special) : 426-434. doi: 10.3934/proc.2001.2001.426

[11]

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

[12]

José Luis Bravo, Manuel Fernández, Armengol Gasull. Stability of singular limit cycles for Abel equations. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1873-1890. doi: 10.3934/dcds.2015.35.1873

[13]

Nikolay Dimitrov. An example of rapid evolution of complex limit cycles. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 709-735. doi: 10.3934/dcds.2011.31.709

[14]

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

[15]

Xiangyu Wang, Laigang Guo. Limit cycles in a switching Liénard system. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022132

[16]

Fabio Scalco Dias, Luis Fernando Mello. The center--focus problem and small amplitude limit cycles in rigid systems. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1627-1637. doi: 10.3934/dcds.2012.32.1627

[17]

Dingheng Pi. Limit cycles for regularized piecewise smooth systems with a switching manifold of codimension two. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 881-905. doi: 10.3934/dcdsb.2018211

[18]

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

[19]

Magdalena Caubergh, Freddy Dumortier, Robert Roussarie. Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle. Communications on Pure and Applied Analysis, 2007, 6 (1) : 1-21. doi: 10.3934/cpaa.2007.6.1

[20]

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

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (261)
  • HTML views (162)
  • Cited by (0)

Other articles
by authors

[Back to Top]