Abstract
This paper deals with the problem of limit cycles for the whirling pendulum equation ẋ = y, ẏ = 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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Baker G L, Blackbuen J A. The Pendulum: A Case Study in Physics. New York: Oxford University Press, 2005
Belley J, Drissi K S. Almost periodic solutions to Josephson’s equation. Nonlinearity, 2003, 16: 35–47
Christopher C, Li C. Limit Cycles of Differential Equations. Berlin: Birkhäuser Verlag, 2007
Gasull A, Geyer A, Mañosas F. On the number of limit cycles for perturbed pendulum equations. J Differential Equations, 2016, 261: 2141–2167
Gasull A, Li C, Torregrosa J. A new Chebyshev family with applications to Abel equations. J Differential Equations, 2012, 252: 1635–1641
Hilbert D. Mathematical problems. Bull Amer Math Soc, 1902, 8: 437–479
Han M. Asymptotic expansions of Melnikov functions and limit cycle bifurcations. Internat J Bifur Chaos Appl Sci Engrg, 2012, 22(12): 1250296
Han M, Li J. Lower bounds for the Hilbert number of polynomial systems. J Differential Equations, 2012, 252: 3278–3304
Han M, Sheng L. Bifurcation of limit cycles in piecewise smooth systems via Melnikov function. J Appl Anal Comput, 2015, 5: 809–815
Inoue K. Perturbed motion of a simple pendulum. J Physical Society of Japan, 1988, 57: 1226–1237
Ilyashenko Y. Centennial history of Hilbert’s 16th problem. Bulletin of the American Mathematical Society, 2022, 39(3): 301–354
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
Jing Z. Chaotic behavior in the Josephson equations with periodic force. SIAM J Appl Math, 1989, 49: 1749–1758
Karlin S, Studden W. Tchebycheff Systems: With Applications in Analysis and Statistic. New York: Interscience Publishers, 1966
Kauderer H. Nichtlineare Mechanik. Berlin: Springer Verlag, 1958
Li C, Zhang Z. A criterion for determining the monotonicity of the ratio of two abelian integrals. J Differential Equations, 1996, 124: 407–424
Liang F, Han M. Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems. Chaos Solitons Fractals, 2012, 45: 454–464
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
Lichardová H. Limit cycles in the equation of whirling pendulum with autonomous perturbation. Appl Math, 1999, 44: 271–288
Lichardová H. The period of a whirling pendulum. Mathematica Bohemica, 2001, 3: 593–606
Liu X, Han M. Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems. Internat J Bifur Chaos Appl Sci Engrg, 2010, 20: 1379–1390
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
Mitrinović D S. Analytic Inequalities. New York: Springer-Verlag, 1970
Mardešić P. Chebyshev Systems and the Versal Unfolding of the Cusp of Order n. Paris: Hermann, 1998
Minorski N. Nonlinear Oscillations. New York: Van Nostrand, 1962
Mawhin J. Global results for the forced pendulum equation//Canada A, Drabek P, Fonda A. Handbook of Differential Equations. Amsterdam: Elsevier, 2004: 533–589
Morozov A D. Limit cycles and chaos in equations of the pendulum type. PMM USSR, 1989, 53: 565–572
Pontryagin L. On dynamical systems close to hamiltonian ones. Zh Exp Theor Phys, 1934, 4: 234–238
Sanders J A, Cushman R. Limit cycles in the Josephson equation. SIAM J Math Anal, 1986, 17: 495–511
Smale S. Mathematical problems for the next century. Math Intell, 1998, 20: 7–15
Teixeira M. Perturbation Theory for Non-Smooth Systems. New York: Springer-Verlag, 2009
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
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
Xiong Y, Han M. Limit cycle bifurcations in a class of perturbed piecewise smooth systems. Applied Mathematics and Computation, 2014, 242: 47–64
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
Yang J. On the number of limit cycles of a pendulum-like equation with two switching lines. Chaos, Solitons and Fractals, 2021, 150: 111092
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
Yang J, Zhao L. Bifurcation of limit cycles of a piecewise smooth Hamiltonian system. Qualitative Theory of Dynamical Systems, 2022, 21: Art 142
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
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest The author declares no conflict of interest.
Additional information
This research was supported by the Natural Science Foundation of Ningxia (2022AAC05044) and the National Natural Science Foundation of China (12161069).
Rights and permissions
About this article
Cite this article
Yang, J. The limit cycle bifurcations of a whirling pendulum with piecewise smooth perturbations. Acta Math Sci 44, 1115–1144 (2024). https://doi.org/10.1007/s10473-024-0319-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-024-0319-4