The nonlinear stability of gas bubbles in acoustic fields is studied using a multiple-scale type of expansion. In particular the development of a subharmonic or a synchronous perturbation to the flow is investigated. It is shown when an equilibrium non-spherical shape oscillation of a bubble is stable. If the amplitude of the sound field is ε then it is shown that subharmonic perturbations of order ε½ can exist and be stable. Furthermore synchronous perturbations of order ε can exist and be stable. It is shown that synchronous perturbations, unlike the subharmonic case where the bifurcation is symmetric, bifurcate transcritically when the driving frequency is varied and also undergo secondary bifurcations. It is further shown that, in certain cases, the latter properties of the synchronous modes cause the flow to exhibit a hysteresis phenomenon when the driving frequency is varied.