The driven pendulum, and E778 tune modulation

The angle θ that a rigid gravity pendulum driven by sinusoidal torque makes with the vertical satisfies d² θ/dt²+(2πQ_I) ² sin(θ)=D cos(2πQ_Mt). The undriven average tune Q_free is Q_I for small angle oscillations, but decreases to zero as the amplitude is increased to π. In general, the driven system behavior depends on D and Q_M, the amplitude and tune of the driving torque. One of four dynamical phases which exist in the (D,Q _M) parameter space exhibits massive chaos. The tune modulation part of the E778 nonlinear dynamics experiment tests the validity of the theoretical models which describe three phases, in an accelerator representation of this extremely simple system. The authors describe the models and present some of the experimental rules. Tune modulation is shown to play an important role in limiting the performance of proton storage rings like the SSC (Superconducting Super Collider)