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The limit cycle bifurcations of a whirling pendulum with piecewise smooth perturbations

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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.

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Authors and Affiliations

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

    Jihua Yang

  2. Ningxia Basic Science Research Center of Mathematics, Yinchuan, 750000, China

    Jihua Yang

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  1. Jihua Yang
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Correspondence to Jihua Yang.

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Conflict of Interest The author declares no conflict of interest.

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This research was supported by the Natural Science Foundation of Ningxia (2022AAC05044) and the National Natural Science Foundation of China (12161069).

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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

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  • DOI: https://doi.org/10.1007/s10473-024-0319-4

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