doi: 10.3934/dcdsb.2021314
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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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 $
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