The order—chaos—order sequence in the spring pendulum

We study the motions of a spring pendulum as a function of its two control parameters (the ratio of the spring and pendulum frequencies, and the energy). It is shown that in the limits for very small and very large parameter values the dynamics of the spring pendulum is predominantly regular, while at intermediate parameter values the majority of initial conditions lead to chaotic trajectories. Thus, upon varying the parameters from small to large values one typically witnesses a transition from order to chaos and back to order again. Similar order—chaos—order sequences are observed in many other dynamical systems, and the spring pendulum is a representative example. In this context, we also discuss the phenomenon for which the spring pendulum is famous, namely the to-and-fro transfer between spring- and pendulum-like behaviour when the spring frequency is (approximately) twice the pendulum frequency. This turns out to play an important role in the order-chaos-order sequence.