Bifurcations in the Mean Angle of a Horizontally Shaken Pendulum: Analysis and Experiment

A pendulum excited by high-frequency horizontal displacement of its pivot point will vibrate with small amplitude about a mean position. The mean value is zero for small excitation amplitudes, but if the excitation is large enough the mean angle can take on non-zero values. This behavior is analyzed using the method of multiple time scales. The change in the mean angle is shown to be the result of a pitchfork bifurcation, or a saddle-node bifurcation if the system is imperfect. Analytical predictions of the mean angle as a function of frequency and amplitude are confirmed by physical experiment and numerical simulation.