  We developed a model to calculate the stability of Gaussian beam
distributions with non-linear space-charge forces in the presence of random and
skew-quadrupole errors. The effect of the space-charge force on the beam matrix
is calculated analytically including full cross-plane coupling in 4D phase
space, which allows us to perform fast parameter studies. For stability
analysis, we find the fixed points of the beam including the space-charge
forces and construct a Jacobi-matrix by slightly perturbing the periodic
solution. The stability of envelope oscillations is inferred by eigenvalue
analysis. Furthermore, we employ envelope tracking as a complementary method
and compare the results of the eigenvalue analysis with FFT data from the
tracked envelope. The non-linearity of the space-charge force in combination
with lattice errors and beam coupling opens up for envelope-lattice resonances
and envelope coupling resonances. Hitting these resonances leads to envelope
blow-up, causing an effective beam mismatch. Therefore, we finally examine the
effect of beam mismatch on the envelope tune-shift and its stability.
