Inverted oscillations of a driven pendulum

Inverted oscillations of a parametrically driven planar pendulum are considered, together with their relationship to the inverted solution. In particular, a horseshoe structure of the associated manifolds is identified which explains the similarity between the bifurcations of the inverted position and the hanging position. This allows us to apply a large body of existing knowledge to the dynamics enabling a lower bound on the forcing required to achieve inverted oscillations to be established.