On the number of limit cycles for perturbed pendulum equations

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• 作者：A. Gasull^a ; ^{gasull@mat.uab.cat" class="auth_mail" title="E-mail the corresponding author} ; A. Geyer^b ; ^{anna.geyer@univie.ac.at" class="auth_mail" title="E-mail the corresponding author} ; F. Mañ ; osas^a ; ^{manyosas@mat.uab.cat" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 34C08 ; secondary ; 34C07 ; 34C25Â ; 37G15 ; 70K05
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 August 2016
• 年：2016
• 卷：261
• 期：3
• 页码：2141-2167
• 全文大小：373 K

文摘

We consider perturbed pendulum-like equations on the cylinder of the form $\ddot{x}+\sin (x)=\epsilon {\sum }_{s=0}^m{Q}_{n,s}(x){\dot{x}}^s$ where Q_n,s$Q_{n,s}$ are trigonometric polynomials of degree n  , and study the number of limit cycles that bifurcate from the periodic orbits of the unperturbed case ε=0$\epsilon =0$ in terms of m and n. Our first result gives upper bounds on the number of zeros of its associated first order Melnikov function, in both the oscillatory and the rotary regions. These upper bounds are obtained expressing the corresponding Abelian integrals in terms of polynomials and the complete elliptic functions of first and second kind. Some further results give sharp bounds on the number of zeros of these integrals by identifying subfamilies which are shown to be Chebyshev systems.