  The space-charge driven envelope instability can be of great danger in high
intensity accelerators and was studied using a two-dimensional (2D) envelope
model and three-dimensional (3D) macroparticle simulations before. In this
paper, we propose a three-dimensional envelope instability model to study the
instability for a bunched beam in a periodic solenoid and radio-frequency (RF)
focusing channel and a periodic quadrupole and RF focusing channel. This study
shows that when the transverse zero current phase advance is below $90$
degrees, the beam envelope can still become unstable if the longitudinal zero
current phase advance is beyond $90$ degrees. For the transverse zero current
phase advance beyond $90$ degrees, the instability stopband width becomes
larger with the increase of the longitudinal focusing strength and even shows
different structure from the 2D case when the longitudinal zero current phase
advance is beyond $90$ degrees. Breaking the symmetry of two longitudinal
focusing RF cavities and the symmetry between the horizontal focusing and the
vertical focusing in the transverse plane in the periodic quadrupole and RF
channel makes the instability stopband broader. This suggests that a more
symmetric accelerator lattice design might help reduce the range of the
envelope instability in parameter space.
