TY - JOUR
T1 - Bifurcation of Limit Cycles of a Perturbed Pendulum Equation
AU - Yang , Jihua
JO - Journal of Nonlinear Modeling and Analysis
VL - 2
SP - 371
EP - 391
PY - 2024
DA - 2024/06
SN - 6
DO - http://doi.org/10.12150/jnma.2024.371
UR - https://global-sci.org/intro/article_detail/jnma/23181.html
KW - Pendulum equation, complete elliptic function, Melnikov function, limit cycle.
AB - <p style="text-align: justify;">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 $n = 1, 2, 3.$</p>