Periodic and subharmonic solutions in the motion of a bead on a rotating circular hoop

Abstract

We establish the necessary conditions for the existence and multiplicity of periodic and subharmonic solutions to a second-order nonlinear ordinary differential equation (ODE). This ODE describes the motion of a bead on a rotating circular hoop subjected to a constant angular velocity $\omega$ and a $T$-periodic forcing. Our approach involves estimating bounds for the angular velocity and period using upper and lower solution methods.