Comparisons between the pendulum with varying length and the pendulum with oscillating support

Abstract

We consider two forced dissipative pendulum systems, the pendulum with vertically oscillating support and the pendulum with periodically varying length, with a view to draw comparisons between their behaviour. We study the two systems for values of the parameters for which the dynamics are non-chaotic. We focus our investigation on the persisting attractive periodic orbits and their basins of attraction, utilising both analytical and numerical techniques. Although in some respect the two systems have similar behaviour, we find that even within the perturbation regime they may exhibit different dynamics. In particular, for the same value of the amplitude of the forcing, the pendulum with varying length turns out to be perturbed to a greater extent. Furthermore the periodic attractors persist under larger values of the damping coefficient in the pendulum with varying length. Finally, unlike the pendulum with oscillating support, the pendulum with varying length cannot be stabilised around the upward position for any values of the parameters.