Computing global orbits of the forced spherical pendulum

Orbits of the spherical pendulum with time-periodic forcing are considered. A numerical framework is developed, which allows orbits to explore the entire “globe”: the spherical pendulum is considered as an invariant manifold in an ambient six-dimensional Euclidean space. The numerical integrator is the second-order Störmer–Verlet method coupled with the Shake–Rattle algorithm. The algorithm preserves numerically the phase space of the sphere, which is a manifold, to machine accuracy. Poincaré sections, restricted to the configuration space, are used to illustrate the transition from oscillatory behavior to chaotic solutions, as the amplitude of the pivot motion is changed. The qualitative change in the Poincaré sections from regular to chaotic behavior is in excellent qualitative agreement with corresponding computations of the Lyapunov exponents (LEs). The LEs are also computed using a novel variant of the Shake–Rattle algorithm. The results show that irregular behavior can explore the entire sphere—even at low forcing amplitudes—and therefore local methods which parameterize only part of the sphere are inadequate in general, and may lead to spurious dynamics. The numerical framework provides a tool for detailed investigation of the symmetric chaos of the forced spherical pendulum.