Year: 2024
Author: Jihua Yang
Journal of Nonlinear Modeling and Analysis, Vol. 6 (2024), Iss. 2 : pp. 371–391
Abstract
This paper investigates the limit cycle bifurcation problem of the pendulum equation on the cylinder of the form $\dot{x} = y, \dot{y} = − {\rm sin} x$ under perturbations of polynomials of ${\rm sin} x,$ ${\rm cos} x$ 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 $n = 1, 2, 3.$
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.12150/jnma.2024.371
Journal of Nonlinear Modeling and Analysis, Vol. 6 (2024), Iss. 2 : pp. 371–391
Published online: 2024-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 21
Keywords: Pendulum equation complete elliptic function Melnikov function limit cycle.