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The Poincaré Bifurcation of a Class of Pendulum Equations

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Abstract:

In this paper, we mainly study the number of limit cycles bifurcate form the periodic orbits of pendulum equations under the perturbations for trigonometric polynomials of degree two. By improving the criterion function of determining the lowest upper bound of the number of zeros of Abelian Integrals, we show that the period annulus (around the origin) can be bifurcate at most two limit cycle (counting multiplicities).

Key words: pendulum equation, poincaré bifurcation, Abelian integral, limit cycle

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Junwen Xu, Hongxing Wu, Yangjian Sun. The Poincaré Bifurcation of a Class of Pendulum Equations[J].Acta mathematica scientia,Series A, 2025, 45(3): 843-849.

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[1]	Cen X, Liu C, Sun Y, Wang J. Cyclicity of period annulus for a class of quadratic reversible systems with a non-rational first integral. Proceedings of the Royal Society of Edinburgh. 2023, 153(5): 1706-1728
[2]	Grau M, Mañosas F Villadelpart J. A Chebychev criterion for Abelian integral. Trans Amer Math Soc, 2011, 363: 109-129
[3]	Gasull A, Geyer A, Mañosas F. On the number of limit cycles for perturbed pendulum equations. J Differ Equ, 2016, 261(3): 2141-2167
[4]	Han M, Yang J. The maximum number of zeros of functions with parameters and application to differential equations. Journal of Nonlinear Modeling and Analysis, 2021, 1: 13-34
[5]	Karlin S, Studden W. Tchebycheff System:With Applications in Analysis and Statistic. New York: Interscience Publishers, 1966
[6]	Li C, Zhang Z. A criterion for determining the monotonocity of the ratio of two Abelian integrals. J Diff Eqns, 1996, 124: 407-424
[7]	Liu C, Xiao D. The smallest upper bound on the number of zeros of Abelian integrals. J Differ Equ, 2020, 269(4): 3816-3852
[8]	Sun Y, Liu C. The Poincaré bifurcation of a SD oscillator. Discrete Contin Dyn Syst Ser B, 2021, 26(3): 1565-1577
[9]	Shi Y, Han M and Zhang L. Homoclinic bifurcationof limit cycles innear Hamiltoniansystems on the cylinder. J Differ Equ, 2021, 304: 1-28

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