Small limit cycles bifurcating in pendulum systems under trigonometric perturbations

In this paper, we consider the bifurcation of small-amplitude limit cycles near the origin in perturbed pendulum systems of the form $\dot{x}=y$, $\dot{y}=-\sin (x)+\epsilon Q(x,y)$, where $Q(x,y)$ is a smooth or piecewise smooth polynomial in the triple $(\sin (x),\cos (x),y)$ with free coefficients. We obtain the sharp upper bound on the number of positive zeros of its associated first order Melnikov function near $h=0$ for $Q(x,y)$ being smooth and piecewise smooth with the discontinuity at $y=0$, respectively.