Subharmonic Frequency Response In A Magnetic Pendulum

We study the subharmonic frequency response of a generalized driven oscillator excited by a nonlinear periodic force. We take a magnetic pendulum called the Doubochinski pendulum as an example. So-called "amplitude quantization," i.e., the existence of multiple discrete periodic solutions, is identified as subharmonic resonance in response to nonlinear feeding. The subharmonic resonance frequency is found to be related to the symmetry of the driving force: Odd subharmonic resonance occurs under an even symmetric driving force, and vice versa. We obtain multiple periodic solutions and investigate the transition and competition between multistable orbits via frequency response curves and Poincare maps. Experimentally observed phenomenon can easily be reproduced in a student laboratory. This provides a perfect example to demonstrate the rich dynamics related to the effect of nonlinear driving within the scope of undergraduate physics.