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September  2021, 14(9): 3167-3181. doi: 10.3934/dcdss.2020337

On the number of limit cycles of a quartic polynomial system

Min Li 1, and  Maoan Han 1,2,,

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  September 2021 Early access  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 $.

Keywords: Quartic polynomial system, limit cycle, Melnikov function.
Mathematics Subject Classification: 37G15, 34C07.
Citation: Min Li, Maoan Han. On the number of limit cycles of a quartic polynomial system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3167-3181. 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

[4]

J. Llibre, Y. 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. Llibre, D. 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. Yang, P. 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. Yu, M. 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. Llibre, Y. 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. Llibre, D. 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. Yang, P. 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. Yu, M. 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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