  A direct photon-phonon parametric effect of the quadratic coupling on the
mean-field dynamics of an optomechanical resonator in the large-scale-movement
regime is found and investigated. Under a weak pumping power, the mechanical
resonator damps to steady state with a nonlinear static response sensitively
modified by the quadratic coupling. When the driving powerincreases beyond the
static energy balance, the steady states lose their stabilities via Hopf
bifurcations and the resonator produces stable self-sustained
oscillation(limit-circle behavior) of discrete energies with step-like
amplitudes due to the parametric effect of the quadratic coupling, which can be
understood roughly by the power balance between gain and loss on the resonator.
A further increase of the pumping power can induce chaotic dynamic of the
resonator via a typical routine of period-doubling bifurcation but which can be
stabilized by the parametric effect through an inversion bifurcation process
back to limit-circle states. The bifurcation-to-inverse-bifurcation transitions
are numerically verified by the maximal Lyapunov exponents of the dynamics and
which indicate an efficient way to suppress the chaotic behavior of the
optomechanical resonator by the quadratic coupling. Furthermore, the parametric
effect of the quadratic coupling on the dynamic transitions of an
optomechanical resonator can be conveniently detected or traced by the output
power spectrum of the cavity field.
