doi: 10.3934/dcdss.2020337

On the number of limit cycles of a quartic polynomial system

1. 

Department of Mathematics, Shanghai Normal University, Shanghai, 200234, PR China

2. 

Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, PR China

* Corresponding author: Maoan Han

Received  March 2019 Revised  October 2019 Published  April 2020

Fund Project: Supported by National Natural Science Foundation of China (11771296 and 11931016)

In this paper, we consider a quartic polynomial differential system with multiple parameters, and obtain the existence and number of limit cycles with the help of the Melnikov function under perturbation of polynomials of degree $ n = 4 $.

Citation: Min Li, Maoan Han. On the number of limit cycles of a quartic polynomial system. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020337
References:
[1]

R. Benterki and J. Llibre, Centers and limit cycles of polynomial differential systems of degree 4 via averaging theory, Journal of Computational and Applied Mathematics, 313 (2017), 273-283.  doi: 10.1016/j.cam.2016.08.047.  Google Scholar

[2]

L. S. Chen and M. S. Wang, The relative position and the number of limit cycles of a quadratic differential system, Acta. Math. Sinica, 22 (1979), 751-758.   Google Scholar

[3]

M. A. Han and Y. Q. Xiong, Limit cycle bifurcations in a class of near-Hamiltonian systems with multiple parameters, Chaos Solitons Fractals, 68 (2014), 20-29.  doi: 10.1016/j.chaos.2014.07.005.  Google Scholar

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J. LlibreY. P. Martínez and C. Valls, Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 887-912.  doi: 10.3934/dcdsb.2018047.  Google Scholar

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J. LlibreD. D. Novaes and M. A. Teixeira, Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearity, 27 (2014), 563-583.  doi: 10.1088/0951-7715/27/3/563.  Google Scholar

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S. L. Shi, A concrete example of the existence of four limit cycles for plane quadratic systems, Sci. Sinica, 23 (1980), 153-158.   Google Scholar

[7]

Y. Tian and P. Yu, Bifurcation of small limit cycles in cubic integrable systems using higher-order analysis, J. Differential Equations, 264 (2018), 5950-5976.  doi: 10.1016/j.jde.2018.01.022.  Google Scholar

[8]

P. Yu and M. A. Han, Four limit cycles from perturbing quadratic integrable systems by quadratic polynomials, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1250254, 28 pp. doi: 10.1142/S0218127412502549.  Google Scholar

[9]

J. M. YangP. Yu and M. A. Han, Limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle of order $m$, Journal of Differential Equations, 266 (2019), 455-492.  doi: 10.1016/j.jde.2018.07.042.  Google Scholar

[10]

P. YuM. Han and Y. Bai, Dynamiocs and bifurcation study on an extended Lorenz system, Journal of Nonlinear Modeling and Analysis, 1 (2019), 107-128.   Google Scholar

show all references

References:
[1]

R. Benterki and J. Llibre, Centers and limit cycles of polynomial differential systems of degree 4 via averaging theory, Journal of Computational and Applied Mathematics, 313 (2017), 273-283.  doi: 10.1016/j.cam.2016.08.047.  Google Scholar

[2]

L. S. Chen and M. S. Wang, The relative position and the number of limit cycles of a quadratic differential system, Acta. Math. Sinica, 22 (1979), 751-758.   Google Scholar

[3]

M. A. Han and Y. Q. Xiong, Limit cycle bifurcations in a class of near-Hamiltonian systems with multiple parameters, Chaos Solitons Fractals, 68 (2014), 20-29.  doi: 10.1016/j.chaos.2014.07.005.  Google Scholar

[4]

J. LlibreY. P. Martínez and C. Valls, Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 887-912.  doi: 10.3934/dcdsb.2018047.  Google Scholar

[5]

J. LlibreD. D. Novaes and M. A. Teixeira, Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearity, 27 (2014), 563-583.  doi: 10.1088/0951-7715/27/3/563.  Google Scholar

[6]

S. L. Shi, A concrete example of the existence of four limit cycles for plane quadratic systems, Sci. Sinica, 23 (1980), 153-158.   Google Scholar

[7]

Y. Tian and P. Yu, Bifurcation of small limit cycles in cubic integrable systems using higher-order analysis, J. Differential Equations, 264 (2018), 5950-5976.  doi: 10.1016/j.jde.2018.01.022.  Google Scholar

[8]

P. Yu and M. A. Han, Four limit cycles from perturbing quadratic integrable systems by quadratic polynomials, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1250254, 28 pp. doi: 10.1142/S0218127412502549.  Google Scholar

[9]

J. M. YangP. Yu and M. A. Han, Limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle of order $m$, Journal of Differential Equations, 266 (2019), 455-492.  doi: 10.1016/j.jde.2018.07.042.  Google Scholar

[10]

P. YuM. Han and Y. Bai, Dynamiocs and bifurcation study on an extended Lorenz system, Journal of Nonlinear Modeling and Analysis, 1 (2019), 107-128.   Google Scholar

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