BREAKDOWN TO CHAOTIC MOTION OF A FORCED, DAMPED, SPHERICAL PENDULUM

The investigation here is of a pendulum forced by a horizontal, sinusoidal, coplanar oscillation of its pivot, moving in nonplanar oscillations near the downward vertical from the pivot. Single or double Fourier expansions of the different forms of periodic solution of the governing differential equations are calculated by Newton's method applied in Fourier space. As the forcing frequency is changed, a typical sequence begins with stable, symmetric, nonplanar oscillations, followed by bifurcation to a stable, periodic modulation of the oscillations, then weakly chaotic modulation merging into fully chaotic modulation of the oscillations. The weakly chaotic oscillations at a given forcing frequency are shown to lie in the neighbourhood of the unstable, strictly periodic oscillatory solutions at the same frequency. Changes in averaged measures such as the Lyapunov exponents are explained in terms of changes in the amount of order in the chaotic oscillations.