
Bifurcation of Limit Cycles of a Perturbed Pendulum Equation 

View Full Text View/Add Comment Download reader 
KeyWord:Pendulum equation, complete elliptic function, Melnikov function, limit cycle 

Hits: 106 
Download times: 122 
Abstract: 
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 antitrigonometric 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 $n=1,2,3$. 


