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We consider perturbed pendulum-like equations on the cylinder of the form $\ddot x+\sin(x)= \varepsilon \sum_{s=0}^{m}{Q_{n,s} (x)\, \dot x^{s}}$ where$Q_{n,s}$ are trigonometric polynomials of degree $n$, and study the number oflimit cycles that bifurcate from the periodic orbits of the unperturbed case$\varepsilon=0$ in terms of $m$ and $n$. Our first result gives upper bounds onthe number of zeros of its associated first order Melnikov function, in boththe oscillatory and the rotary regions. These upper bounds are obtainedexpressing the corresponding Abelian integrals in terms of polynomials and thecomplete elliptic functions of first and second kind. Some further results givesharp bounds on the number of zeros of these integrals by identifyingsubfamilies which are shown to be Chebyshev systems.

On the number of limit cycles for perturbed pendulum equations