On the number of limit cycles of a quartic polynomial system

Min Li; Maoan Han

doi:10.3934/dcdss.2020337

Article Contents

2021, Volume 14, Issue 9: 3167-3181. Doi: 10.3934/dcdss.2020337

This issue Previous Article Uniform polynomial stability of second order integro-differential equations in Hilbert spaces with positive definite kernels Next Article Positive solutions of the discrete Robin problem with $ \phi $-Laplacian

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
^* Corresponding author: Maoan Han

Received: February 28, 2019

Revised: September 30, 2019

Early access: April 2020

Published: September 2021

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

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Abstract

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 $.

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Mathematics Subject Classification: 37G15, 34C07.

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