  We study the classical dynamics of a membrane inside a cavity in the
situation where this optomechanical system possesses a reflection symmetry.
Symmetry breaking occurs through supercritical and subcritical pitchfork
bifurcations of the static fixed point solutions. Both bifurcations can be
observed through variation of the laser-cavity detuning, which gives rise to a
boomerang-like fixed point pattern with hysteresis. The symmetry-breaking fixed
points evolve into self-sustained oscillations when the laser intensity is
increased. In addition to the analysis of the accompanying Hopf bifurcations we
describe these oscillations at finite amplitudes with an ansatz that fully
accounts for the frequency shift relative to the natural membrane frequency. We
complete our study by following the route to chaos for the membrane dynamics.
