THE ORDER-CHAOS-ORDER SEQUENCE IN THE SPRING PENDULUM

We study the motions of a spring pendulum as a function of its two con trol parameters (the ratio of the spring and pendulum frequencies, and the energy). It is shown that in the limits for very small and very l arge parameter values the dynamics of the spring pendulum is predomina ntly regular, while at intermediate parameter values the majority of i nitial conditions lead to chaotic trajectories. Thus, upon varying the parameters from small to large values one typically witnesses a trans ition 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, namel y the to-and-fro transfer between spring- and pendulum-like behaviour when the spring frequency is (approximately) twice the pendulum freque ncy. This turns out to play an important role in the order-chaos-order sequence.