  The widespread phenomena of multistability is a problem involving rich
dynamics to be explored. In this paper, we study the multistability of a
generalized nonlinear forcing oscillator excited by $f(x)cos \omega t$. We take
Doubochinski's Pendulum as an example. The so-called "amplitude quantization",
i.e., the multiple discrete periodical solutions, is identified as
self-adaptive subharmonic resonance in response to nonlinear feeding. The
subharmonic resonance frequency is found related to the symmetry of the driving
force: odd subharmonic resonance occurs under even symmetric driving force and
vice versa. We solve the multiple periodical solutions and investigate the
transition and competition between these multi-stable modes via frequency
response curves and Poincare maps. We find the irreversible transition between
the multistable modes and propose a multistability control strategy.
