On the number of limit cycles of a pendulum-like equation with two switching lines

Abstract

This paper is devoted to study the limit cycle bifurcations of a pendulum equation x˙=y,y˙=−sinx under non-smooth perturbations of polynomials of cosx, sinx and y of degree n with switching lines x=0 and y=0. The upper bounds of the number of limit cycles in both the oscillatory and the rotary regions are obtained by expressing the corresponding first order Melnikov functions as several generating functions, some of which are complete elliptic integrals of the first and second kind.