A mathematical model describing dynamics of the cluster of gas bubbles in an acoustic field is presented. According to this model a cluster is considered as a large drop with microbubbles inside. The proposed model is used as a basis (1) for an analytical study of small bubble oscillations in mono- and polydisperse clusters and (2) for numerical investigations of nonlinear bubble oscillations and of the diffusion stability of gas bubbles in the cluster. A synchronization of the collapse phases of bubbles with different radii and collapse intensification for bubbles of one size in the presence of bubbles of other size is found. These effects are explained by the interaction between the bubbles of different radii in the cluster. For the cluster with one radius bubbles the numerical values are obtained for the initial gas concentrations in the liquid at which the bubbles tend to one of two equilibrium states because of rectified diffusion. It is found that the cluster with the bubbles of two different radii tends to become a cluster with the bubbles of one radius due to rectified diffusion.