具有尖点环的非光滑微分系统的极限环分支

数学物理学报 ›› 2023, Vol. 43 ›› Issue (6): 1667-1680.

具有尖点环的非光滑微分系统的极限环分支

杨纪华^*(),马亮

宁夏师范学院数学与计算机科学学院宁夏固原 756000

收稿日期:2022-11-22 修回日期:2023-05-17 出版日期:2023-12-26 发布日期:2023-11-16
通讯作者: *杨纪华，E-mail: jihua1113@163.com
基金资助:
国家自然科学基金(12161069);宁夏自然科学基金(2022AAC05044);宁夏高等学校一流学科建设(教育学学科)(NXYLXK2021B10)

Limit Cycle Bifurcations of a Non-smooth Differential System with a Cuspidal Loop

Yang Jihua^*(),Ma Liang

School of Mathematics and Computer Science, Ningxia Normal University, Ningxia Guyuan 756000

Received:2022-11-22 Revised:2023-05-17 Online:2023-12-26 Published:2023-11-16
Supported by:
NSFC(12161069);Natural Science Foundation of Ningxia(2022AAC05044);Construction of First-class Disciplines of Higher Education of Ningxia (pedagogy)(NXYLXK2021B10)

摘要/Abstract

摘要：

研究具有尖点环的非光滑微分系统在 $n$ 次多项式非光滑扰动下的极限环分支问题. 首先把扰动微分系统的一阶 Melnikov 函数 $M(h)$ 表示成几个具有多项式系数的生成积分的线性组合, 并用数学归纳法证明这些多项式的系数是相互独立的常数. 然后应用 $M(h)$ 的渐近展式得到从原点和尖点环附近分支出极限环个数的下界.

关键词: 极限环, Melnikov 函数, 尖点环, 渐近展式

Abstract:

This paper studies the limit cycle bifurcation problem of a non-smooth differential system with a cuspidal loop under non-smooth perturbation of polynomials of degree $n$. Firstly, the first order Melnikov function $M(h)$ of the perturbed differential system is expressed as a linear combination of several generating integrals with polynomial coefficients, and the independence of coefficients of these polynomials is proved by mathematical induction. Then the lower bounds of the number of limit cycles bifurcating the origin and cuspidal loop are obtained by using the asymptotic expansions of $M(h)$.

Key words: Limit cycle, Melnikov function, Cuspidal loop, Asymptotic expansion

中图分类号:

O175.12

杨纪华, 马亮. 具有尖点环的非光滑微分系统的极限环分支[J]. 数学物理学报, 2023, 43(6): 1667-1680.

Yang Jihua, Ma Liang. Limit Cycle Bifurcations of a Non-smooth Differential System with a Cuspidal Loop[J]. Acta mathematica scientia,Series A, 2023, 43(6): 1667-1680.

图/表 1

参考文献 16

[1]	Tse C, Bernardo M. Complex behavior in switching power converter. Proceeding of IEEE, 2002, 90(5): 768-781 doi: 10.1109/JPROC.2002.1015006
[2]	Kember S, Babitsky V. Excitation of vibro-impact systems by periodic impulses. J Sound Vib, 1999, 227(2): 427-447 doi: 10.1006/jsvi.1999.2353
[3]	黄立宏, 郭振远, 王佳伏. 右端不连续微分方程理论与应用. 北京: 科学出版社, 2011
	Huang L H, Guo Z Y, Wang F J. Theory and Application of Differential Equations with Discontinuous Right-Hand Sides. Beijing: Science Press, 2011
[4]	Zou Y, Kupper T, Beyn W. Generalized Hopf bifurcation for planar Filippov systems continuous at the origin. J Nonlinear Sci, 2006, 16(2): 159-177 doi: 10.1007/s00332-005-0606-8
[5]	Simpson D, Meiss J. Andronov-Hopf bifurcation in planar, piecewise-smooth, continuous flows. Phys Lett A, 2007, 371(3): 213-220 doi: 10.1016/j.physleta.2007.06.046
[6]	Virgin L, Begley C. Grazing bifurcations and basins of attraction in an impact-friction oscillator. Physica D, 1999, 130: 43-57 doi: 10.1016/S0167-2789(99)00016-0
[7]	Kukučka P. Melnikov method for discontinuous planar systems. Nonlinear Anal, 2007, 66: 2698-2719 doi: 10.1016/j.na.2006.04.001
[8]	Li L, Huang L. Concurrent homoclinic bifurcation and Hopf bifurcation for a class of planar Filippov systems. J Math Anal Appl, 2014, 411: 83-94 doi: 10.1016/j.jmaa.2013.09.025
[9]	Liu X, Han M A. Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems. Int J Bifurc Chaos, 2010, 20(5): 1379-1390 doi: 10.1142/S021812741002654X
[10]	Llibre J, Lopes B, Moraes J. Limit cycles for a class of continuous and discontinuous cubic polynomial differential systems. Qual Theory Dyn Syst, 2014, 13: 129-148 doi: 10.1007/s12346-014-0109-9
[11]	Li F, Yu P, Tian Y, Liu Y. Center and isochronous center conditions for switching systems associated with elementary singular points. Commun Nonlinear Sci Numer Simul, 2015, 28: 81-97 doi: 10.1016/j.cnsns.2015.04.005
[12]	Guo L, Yu P, Chen Y. Bifurcation analysis on a class of $Z_2$-equivariant cubic switching systems showing eighteen limit cycles. J Differential Equations, 2019, 266: 1221-1244 doi: 10.1016/j.jde.2018.07.071
[13]	Liang F, Han M. Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems. Chaos Solitons Fractals, 2012, 45: 454-464 doi: 10.1016/j.chaos.2011.09.013
[14]	Liang F, Han M, Zhang X. Bifurcation of limit cycles from generalized homoclinic loops in planar piecewise smooth systems. J Differential Equations, 2013, 255: 4403-4436 doi: 10.1016/j.jde.2013.08.013
[15]	Xiong Y, Han M. Limit cycles appearing from a generalized heteroclinic loop with a cusp and a nilpotent saddle. J Differential Equations, 2021, 303: 575-607 doi: 10.1016/j.jde.2021.09.031
[16]	Han M, Sheng L. Bifurcation of limit cycles in piecewise smooth systems via Melnikov function. J Appl Anal Comput, 2015, 5: 809-815