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A direct photon-phonon parametric effect of the quadratic coupling on themean-field dynamics of an optomechanical resonator in the large-scale-movementregime is found and investigated. Under a weak pumping power, the mechanicalresonator damps to steady state with a nonlinear static response sensitivelymodified by the quadratic coupling. When the driving powerincreases beyond thestatic energy balance, the steady states lose their stabilities via Hopfbifurcations and the resonator produces stable self-sustainedoscillation(limit-circle behavior) of discrete energies with step-likeamplitudes due to the parametric effect of the quadratic coupling, which can beunderstood roughly by the power balance between gain and loss on the resonator.A further increase of the pumping power can induce chaotic dynamic of theresonator via a typical routine of period-doubling bifurcation but which can bestabilized by the parametric effect through an inversion bifurcation processback to limit-circle states. The bifurcation-to-inverse-bifurcation transitionsare numerically verified by the maximal Lyapunov exponents of the dynamics andwhich indicate an efficient way to suppress the chaotic behavior of theoptomechanical resonator by the quadratic coupling. Furthermore, the parametriceffect of the quadratic coupling on the dynamic transitions of anoptomechanical resonator can be conveniently detected or traced by the outputpower spectrum of the cavity field.

Photon-phonon parametric oscillation induced by the quadratic coupling in an optomechanical resonator