Adaptive resonance and pumping a swing

Resonance, and particularly parametric resonance, is often given as an explanation for the increase in amplitude of a child on a swing, or more generally the actuation of pendulums. However the analysis is based on linearized models of the pendulum. We revisit the definition of parametric resonance and show how the pumping phenomenon is a much more complicated process than standard accounts might suggest. We show that if the frequency is not adapted as a function of the period of the oscillations then parametric forcing leads to a modulated oscillation which grows and then decreases periodically. A weakly nonlinear approximation confirms this observation. This shows that the parametric forcing required to maintain or grow oscillations in an actuated pendulum need to adapt to the period of the current oscillations rather than some abstract `natural frequency'. These results are extended to other resonant models of pumping a swing.