Bifurcation of Limit Cycles of a Perturbed Pendulum Equation

This paper investigates the limit cycle bifurcation problem of the pendulum equation on the cylinder of the form $\dot{x}=y, \dot{y}=-\sin x$ under perturbations of polynomials of $\sin x$, $\cos x$ and $y$ of degree $n$ with a switching line $y=0$. We first prove that the corresponding first order Melnikov functions can be expressed as combinations of anti-trigonometric functions and the complete elliptic functions of first and second kind with polynomial coefficients in both the oscillatory and rotary regions for arbitrary $n$. We also verify the independence of coefficients of these polynomials. Then, the lower bounds for the number of limit cycles are obtained using their asymptotic expansions with $n=1,2,3$.