The phonon spectra, distortion mechanisms, and thermoelastic properties of CaSiO3 and MgSiO3 in the cubic perovskite structure are investigated as a function of pressure using a lattice dynamic approach. The bonding forces are derived from a parameter-free rigid-ion electron-gas formulation. At low pressures, CaSiO3 is found to be dynamically stable in the cubic perovskite structure; however, the phonon spectrum exhibits soft modes at the Brillouin zone boundary which ultimately result in a dynamic instability of the lattice near 80 GPa. The computed phonon spectrum of cubic MgSiO3 perovskite exhibits complex frequencies along parts of the zone boundary at all densities investigated. These vibrational instabilities include coupled octahedral rotations which produce the observed distorted structure of MgSiO3 perovskite. The measured bulk modulus of MgSiO3 perovskite compares well with our calculated value for the cubic structure.