(untitled) (Journal of Nonlinear Modeling and Analysis, 2024)
This paper investigates the limit cycle bifurcation problem of the pendulum equation on the cylinder of the form x = y, 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.