-
Previous Article
Positive solutions of the discrete Robin problem with $ \phi $-Laplacian
- DCDS-S Home
- This Issue
-
Next Article
Uniform polynomial stability of second order integro-differential equations in Hilbert spaces with positive definite kernels
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.
|
[1] |
Yulin Zhao, Siming Zhu. Higher order Melnikov function for a quartic hamiltonian with cuspidal loop. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 995-1018. doi: 10.3934/dcds.2002.8.995 |
[2] |
Valery A. Gaiko. The geometry of limit cycle bifurcations in polynomial dynamical systems. Conference Publications, 2011, 2011 (Special) : 447-456. doi: 10.3934/proc.2011.2011.447 |
[3] |
Hang Zheng, Yonghui Xia. Chaotic threshold of a class of hybrid piecewise-smooth system by an impulsive effect via Melnikov-type function. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2021319 |
[4] |
Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047 |
[5] |
Sze-Bi Hsu, Junping Shi. Relaxation oscillation profile of limit cycle in predator-prey system. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 893-911. doi: 10.3934/dcdsb.2009.11.893 |
[6] |
David Yang Gao, Changzhi Wu. On the triality theory for a quartic polynomial optimization problem. Journal of Industrial and Management Optimization, 2012, 8 (1) : 229-242. doi: 10.3934/jimo.2012.8.229 |
[7] |
Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114 |
[8] |
Kuan-Ju Huang, Yi-Jung Lee, Tzung-Shin Yeh. Classification of bifurcation curves of positive solutions for a nonpositone problem with a quartic polynomial. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1497-1514. doi: 10.3934/cpaa.2016.15.1497 |
[9] |
Jackson Itikawa, Jaume Llibre. Global phase portraits of uniform isochronous centers with quartic homogeneous polynomial nonlinearities. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 121-131. doi: 10.3934/dcdsb.2016.21.121 |
[10] |
Qiaoyi Hu, Zhijun Qiao. Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6975-7000. doi: 10.3934/dcds.2016103 |
[11] |
Hassan Emamirad, Philippe Rogeon. Semiclassical limit of Husimi function. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 669-676. doi: 10.3934/dcdss.2013.6.669 |
[12] |
Ben Niu, Weihua Jiang. Dynamics of a limit cycle oscillator with extended delay feedback. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1439-1458. doi: 10.3934/dcdsb.2013.18.1439 |
[13] |
Magdalena Caubergh, Freddy Dumortier, Robert Roussarie. Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle. Communications on Pure and Applied Analysis, 2007, 6 (1) : 1-21. doi: 10.3934/cpaa.2007.6.1 |
[14] |
Jihua Yang, Liqin Zhao. Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2417-2425. doi: 10.3934/dcdsb.2017123 |
[15] |
Stijn Luca, Freddy Dumortier, Magdalena Caubergh, Robert Roussarie. Detecting alien limit cycles near a Hamiltonian 2-saddle cycle. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1081-1108. doi: 10.3934/dcds.2009.25.1081 |
[16] |
Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure and Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257 |
[17] |
Jaume Llibre, Claudia Valls. Algebraic limit cycles for quadratic polynomial differential systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2475-2485. doi: 10.3934/dcdsb.2018070 |
[18] |
Fang Wu, Lihong Huang, Jiafu Wang. Bifurcation of the critical crossing cycle in a planar piecewise smooth system with two zones. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021264 |
[19] |
Yulin Zhao. On the monotonicity of the period function of a quadratic system. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 795-810. doi: 10.3934/dcds.2005.13.795 |
[20] |
Jaume Llibre, Yilei Tang. Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1769-1784. doi: 10.3934/dcdsb.2018236 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]