THE LIMIT CYCLE BIFURCATIONS OF A WHIRLING PENDULUM WITH PIECEWISE SMOOTH PERTURBATIONS

doi:10.1007/s10473-024-0319-4

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (3): 1115-1144.doi: 10.1007/s10473-024-0319-4

THE LIMIT CYCLE BIFURCATIONS OF A WHIRLING PENDULUM WITH PIECEWISE SMOOTH PERTURBATIONS

Jihua yang^1,2

1. School of Mathematics and Computer Science, Ningxia Normal University, Guyuan 756000, China;
2. Ningxia Basic Science Research Center of Mathematics, Yinchuan 750000, China

收稿日期:2022-10-18 修回日期:2023-03-24 出版日期:2024-06-25 发布日期:2024-05-21

THE LIMIT CYCLE BIFURCATIONS OF A WHIRLING PENDULUM WITH PIECEWISE SMOOTH PERTURBATIONS

Jihua yang^1,2

1. School of Mathematics and Computer Science, Ningxia Normal University, Guyuan 756000, China;
2. Ningxia Basic Science Research Center of Mathematics, Yinchuan 750000, China

Received:2022-10-18 Revised:2023-03-24 Online:2024-06-25 Published:2024-05-21
About author:Jihua yang,E-mail:yangjh@mail.bnu.edu.cn; jihua1113@163.com
Supported by:
Natural Science Foundation of Ningxia (2022AAC05044) and the National Natural Science Foundation of China (12161069).

摘要/Abstract

摘要： This paper deals with the problem of limit cycles for the whirling pendulum equation $\dot{x}=y,\ \dot{y}=\sin x(\cos x-r)$ under piecewise smooth perturbations of polynomials of $\cos x$, $\sin x$ and $y$ of degree $n$ with the switching line $x=0$. The upper bounds of the number of limit cycles in both the oscillatory and the rotary regions are obtained using the Picard-Fuchs equations, which the generating functions of the associated first order Melnikov functions satisfy. Furthermore, the exact bound of a special case is given using the Chebyshev system. At the end, some numerical simulations are given to illustrate the existence of limit cycles.

关键词: whirling pendulum, limit cycle, Melnikov function, Picard-Fuchs equation, Chebyshev system

Abstract: This paper deals with the problem of limit cycles for the whirling pendulum equation $\dot{x}=y,\ \dot{y}=\sin x(\cos x-r)$ under piecewise smooth perturbations of polynomials of $\cos x$, $\sin x$ and $y$ of degree $n$ with the switching line $x=0$. The upper bounds of the number of limit cycles in both the oscillatory and the rotary regions are obtained using the Picard-Fuchs equations, which the generating functions of the associated first order Melnikov functions satisfy. Furthermore, the exact bound of a special case is given using the Chebyshev system. At the end, some numerical simulations are given to illustrate the existence of limit cycles.

Key words: whirling pendulum, limit cycle, Melnikov function, Picard-Fuchs equation, Chebyshev system

中图分类号:

34C08

Jihua yang. THE LIMIT CYCLE BIFURCATIONS OF A WHIRLING PENDULUM WITH PIECEWISE SMOOTH PERTURBATIONS[J]. 数学物理学报(英文版), 2024, 44(3): 1115-1144.

Jihua yang. THE LIMIT CYCLE BIFURCATIONS OF A WHIRLING PENDULUM WITH PIECEWISE SMOOTH PERTURBATIONS[J]. Acta mathematica scientia,Series B, 2024, 44(3): 1115-1144.

参考文献

[1] Baker G L, Blackbuen J A.The Pendulum: A Case Study in Physics. New York: Oxford University Press, 2005
[2] Belley J, Drissi K S. Almost periodic solutions to Josephson's equation. Nonlinearity, 2003, 16: 35-47
[3] Christopher C, Li C.Limit Cycles of Differential Equations. Berlin: Birkhäuser Verlag, 2007
[4] Gasull A, Geyer A,Mañosas F. On the number of limit cycles for perturbed pendulum equations. J Differential Equations, 2016, 261: 2141-2167
[5] Gasull A, Li C, Torregrosa J. A new Chebyshev family with applications to Abel equations. J Differential Equations, 2012, 252: 1635-1641
[6] Hilbert D. Mathematical problems. Bull Amer Math Soc, 1902, 8: 437-479
[7] Han M. Asymptotic expansions of Melnikov functions and limit cycle bifurcations. Internat J Bifur Chaos Appl Sci Engrg, 2012, 22(12): 1250296
[8] Han M, Li J. Lower bounds for the Hilbert number of polynomial systems. J Differential Equations, 2012, 252: 3278-3304
[9] Han M, Sheng L. Bifurcation of limit cycles in piecewise smooth systems via Melnikov function. J Appl Anal Comput, 2015, 5: 809-815
[10] Inoue K. Perturbed motion of a simple pendulum. J Physical Society of Japan, 1988, 57: 1226-1237
[11] Ilyashenko Y. Centennial history of Hilbert's 16th problem. Bulletin of the American Mathematical Society, 2022, 39(3): 301-354
[12] Jing Z, Cao H. Bifurcations of periodic orbits in a Josephson equation with a phase shift. International Journal of Bifurcation and Chaos, 2002, 12: 1515-1530
[13] Jing Z. Chaotic behavior in the Josephson equations with periodic force. SIAM J Appl Math, 1989, 49: 1749-1758
[14] Karlin S, Studden W.Tchebycheff Systems: With Applications in Analysis and Statistic. New York: Interscience Publishers, 1966
[15] Kauderer H. Nichtlineare Mechanik. Berlin: Springer Verlag, 1958
[16] Li C, Zhang Z. A criterion for determining the monotonicity of the ratio of two abelian integrals. J Differential Equations, 1996, 124: 407-424
[17] Liang F, Han M. Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems. Chaos Solitons Fractals, 2012, 45: 454-464
[18] Liang F, Han M, Romanovski V. Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop. Nonlinear Anal, 2012, 75: 4355-4374
[19] Lichardová H. Limit cycles in the equation of whirling pendulum with autonomous perturbation. Appl Math, 1999, 44: 271-288
[20] Lichardová H. The period of a whirling pendulum. Mathematica Bohemica, 2001, 3: 593-606
[21] Liu X, Han M. Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems. Internat J Bifur Chaos Appl Sci Engrg, 2010, 20: 1379-1390
[22] Llibre J, Ramírez R, Ramírez V, Sadovskaia N. The 16th Hilbert problem restricted to circular algebraic limit cycles. J Differ Equ, 2016, 260: 5726-5760
[23] Mitrinović D S. Analytic Inequalities.New York: Springer-Verlag, 1970
[24] Mardešić P. Chebyshev Systems and the Versal Unfolding of the Cusp of Order $n$. Paris: Hermann, 1998
[25] Minorski N. Nonlinear Oscillations. New York: Van Nostrand, 1962
[26] Mawhin J. Global results for the forced pendulum equation//Canada A, Drabek P, Fonda A. Handbook of Differential Equations. Amsterdam: Elsevier, 2004: 533-589
[27] Morozov A D. Limit cycles and chaos in equations of the pendulum type. PMM USSR, 1989, 53: 565-572
[28] Pontryagin L. On dynamical systems close to hamiltonian ones. Zh Exp Theor Phys, 1934, 4: 234-238
[29] Sanders J A, Cushman R. Limit cycles in the Josephson equation. SIAM J Math Anal, 1986, 17: 495-511
[30] Smale S. Mathematical problems for the next century. Math Intell, 1998, 20: 7-15
[31] Teixeira M.Perturbation Theory for Non-Smooth Systems. New York: Springer-Verlag, 2009
[32] Wang N, Wang J, Xiao D. The exact bounds on the number of zeros of complete hyperelliptic integrals of the first kind. J Differential Equations, 2013, 254: 323-341
[33] Wei L, Zhang X. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete and Continuous Dynamical Systems, 2016, 36(5): 2803-2825
[34] Xiong Y, Han M. Limit cycle bifurcations in a class of perturbed piecewise smooth systems. Applied Mathematics and Computation, 2014, 242: 47-64
[35] Xiong Y, Han M, Romanovski V G. The maximal number of limit cycles in perturbations of piecewise linear Hamiltonian systems with two saddles. Internat J Bifur Chaos, 2017, 27(8): 1750126
[36] Yang J. On the number of limit cycles of a pendulum-like equation with two switching lines. Chaos, Solitons and Fractals, 2021, 150: 111092
[37] Yang J, Zhang E. On the number of limit cycles for a class of piecewise smooth Hamiltonian systems with discontinuous perturbations. Nonlinear Analysis: Real World Applications, 2020, 52: 103046
[38] Yang J, Zhao L. Bifurcation of limit cycles of a piecewise smooth Hamiltonian system. Qualitative Theory of Dynamical Systems, 2022, 21: Art 142
[39] Zhao Y, Zhang Z. Linear estimate of the number of zeros of Abelian integrals for a kind of quartic Hamiltonians. J Differential Equations, 1999, 155: 73-88

THE LIMIT CYCLE BIFURCATIONS OF A WHIRLING PENDULUM WITH PIECEWISE SMOOTH PERTURBATIONS

THE LIMIT CYCLE BIFURCATIONS OF A WHIRLING PENDULUM WITH PIECEWISE SMOOTH PERTURBATIONS

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 7

Metrics

本文评价

推荐阅读 10

[1]	Bo HUANG, Wei NIU, Dongming WANG. SYMBOLIC COMPUTATION FOR THE QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS[J]. 数学物理学报(英文版), 2022, 42(6): 2478-2504.
[2]	梁峰, 韩茂安, 江潮源. LIMIT CYCLE BIFURCATIONS OF A PLANAR NEAR-INTEGRABLE SYSTEM WITH TWO SMALL PARAMETERS[J]. 数学物理学报(英文版), 2021, 41(4): 1034-1056.
[3]	尚德生; 张同华. BIFURCATIONS OF A CUBIC SYSTEM PERTURBED BY DEGREE FIVE[J]. 数学物理学报(英文版), 2009, 29(1): 11-24.
[4]	刘振海; E. Sáez; I. Szántó. LIMIT CYCLES AND INVARIANT PARABOLA IN A KUKLES SYSTEM OF DEGREE THREE[J]. 数学物理学报(英文版), 2008, 28(4): 865-869.
[5]	丰建文, 陈士华. A CLASS OF QUADRATIC HAMILTONIAN SYSTEMS UNDER QUADRATIC PERTURBATION[J]. 数学物理学报(英文版), 2001, 21(2): 249-253.
[6]	黄立宏, 陈明博. BOUNDEDNESS OF SOLUTIONS AND EXISTENCE OF LIMIT CYCLES FOR A NONLINEAR SYSTEM OF DIFFERENTIAL EQUATIONS[J]. 数学物理学报(英文版), 1998, 18(1): 113-120.
[7]	曾唯尧, 钱祥征, 王志成. EXPONENTIAL DICHOTOMIES,HOMOCLINIC ORBITS AND METHODS OF MELNIKOV[J]. 数学物理学报(英文版), 1996, 16(1): 113-120.