  In the framework of the nonsecular perturbation theory based on the
Bogoliubov averaging method, an optomechanical system with an asymmetric
anharmonic mechanical resonator is studied. The cross-Kerr interaction and the
Kerr-like self-interaction of photons and vibration quanta arise in the
Hamiltonian. These interactions are induced by both cubic and quartic
nonlinearities of oscillations of the mechanical resonator and the
cavity-resonator interaction that is linear in mechanical displacements. We
demonstrate a bistable behavior of the number of vibration quanta and find that
this behavior is controlled by the cross-Kerr interaction. It is shown that,
without driving and dissipation, the constructed superposition
Yurke-Stoler-like states of the cavity (or the mechanical resonator)
disentangle at certain times the entangled modes of the system. The obtained
results offer new possibilities for control of optomechanical systems with
asymmetric mechanical oscillations.
