New Dynamical Aspects of the Driven Pendulum

The dynamics of the driven pendulum have been studied numerically and analytically under non-resonance condition, where the driving frequency is far from the characteristic frequency of the pendulum. It is found that, as is known previously, a symmetry·breaking occurs when the driving amplitude j exceeds a critical value, but further increase of j does not lead to a chaotic phase of motion, but to an inversion of the pendulum. Namely, in a definite range of j, the pendulum oscillates in the inverted configuration. The normal and inverted configurations appear alternately as j still increases. It is also found that, for intermediate values of j, at most four phases of motion (single-period as well as period-tripled oscillations, and modulated rotations in the positive or negative direction) can coexist with each other. Most of the dynamical structures obtained numeri cally can be explained by perturbation expansion solutions of the equation of motion.