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In this paper, we consider the bifurcation of small-amplitude limit cyclesnear the origin in perturbed pendulum systems of the form $\dot x= y$, $\doty=-\sin(x)+\varepsilon Q(x,y)$, where $Q(x,y)$ is a smooth or piecewise smoothpolynomial in the triple $(\sin(x),\cos(x), y)$ with free coefficients. Weobtain the sharp upper bound on the number of positive zeros of its associatedfirst order Melnikov function near $h=0$ for $Q(x,y)$ being smooth andpiecewise smooth with the discontinuity at $y=0$, respectively.

Small limit cycles bifurcating in pendulum systems under trigonometric perturbations