Linear beam-beam resonance due to coherent dipole motion

We study the transverse motion of the barycentres of bunches in two beams, which circulate in opposite direction in a storage ring, and are coupled by the beam-beam effect. the motion is described by a linear system of escillators, and represented by a matrix. We distinguish between perfect machines in which the bunch parameters and the arc and interaction point parameters are all equal, and machines with errors in which at least one of these conditions is not satisfied. We determine the regions in tune space where the motion is unstable, analytically for perfect machines, and numerically for machines with errors. We identify these regions as resonances related to the tunes of one of the two beams or to their sum. We establish what resonances are excited under given conditions. We find that more resonances occur in machines with errors than in perfect machines. by multi-particle tracking, we study the instability at finite amplitudes.