Nonlinear resonance for a generalized parametric oscillator

Multiple discrete stationary solutions, namely “amplitude quantization”, found in kick-excited pendulums have not been well understood over decades. We show that these discrete solutions are subharmonic resonance originated from the nonlinear
periodic driving force. From theoretical analysis, we reveal the relationship between subharmonic resonance frequency and
symmetry of the driving force: odd subharmonic resonance occurs under even symmetric driving force and vice versa. We also
show that multiple periodic solutions coexist near subharmonic resonance frequencies, in particular dual solutions are
discovered. While the usual parametric oscillator has periodic driving force proportional to displacement and experience
subharmonic resonance at even multiples of the pendulum's frequency, our mathematical model can be viewed as generalized
parametric resonance which has nonlinear periodic driving force and experience subharmonic resonance at integral multiples
of the pendulum's frequency.
In order to investigate the stability conditions and evolution of the solutions, we calculate the frequency response
curves, bifurcation diagram, phase diagrams, Poincare maps and the stability diagrams.