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On the number of limit cycles of a quartic polynomial system

  • * Corresponding author: Maoan Han

    * Corresponding author: Maoan Han
Supported by National Natural Science Foundation of China (11771296 and 11931016)
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  • 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 $.

    Mathematics Subject Classification: 37G15, 34C07.

    Citation:

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  • [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. 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.
    [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.
    [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. 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.
    [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. 
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