On the Existence and Stability of Periodic Solutions for a Bead on a Rotating Circular Hoop

This paper investigates sufficient conditions for the existence of odd periodic solutions in periodically forced Newtonian differential equations, specifically addressing cases where certain hypotheses from recent literature are not met. We build upon a known existence result of R. Ortega for oscillators with Dirichlet-type boundary conditions, relaxing one of the assumptions required therein. Unlike Ortega’s original result, our derived conditions are solely sufficient but offer enhanced applicability due to their simplified nature. As an application, we analyze the existence of odd periodic solutions for a Bead on a Rotating Circular Hoop, a classical model in mechanics, discussing the compatibility of the approach with the stability properties of the lower equilibrium. We establish conditions under which the lower equilibrium exhibits a twist-type behavior, thereby implying its stability. Furthermore, for this specific model, we address the existence and stability of m-subharmonic solutions of constant sign (lateral oscillations), combining the method of lower and upper solutions with recent results on twist-type dynamics and KAM theory.