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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 |
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 $.
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.
|
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. |
| [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.
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