In the case of proton synchrotrons the instability is always followed by turbulent bunch lengthening, but in the case of electron storage rings where radiation damping causes the particles to be confined, the behavior of the beam beyond the instability threshold may vary depending on the shape of the wake fields and the intensity. The signature of this behavior is strong quadrupole oscillations. Another feature is that as intensity increases, low-frequency sidebands begin to appear around the quadrupole frequency. This qualitatively resembles an effect observed at LEP in 1991. It is common to study stability of a beam from the point of view of the stability of a stationary distribution with respect to an infinitesimal perturbation.
Questions remain about what happens to the beam when the threshold intensity is exceeded.
In the case of protons, a turbulent bunch lengthening effect is observed, i.e. the bunch distribution alters under the self forces until the distribution becomes stable. At the threshold intensity the bunch becomes unstable, but radiation damping causes the particles to be confined and the instability does not necessarily cause loss of particles. The nonlinear regime is extremely difficult to treat analytically. We used a multiparticle tracking technique as described in this paper. The radiation damping time t e was set to 500 - 2,000 turns and the other parameters have been chosen to obtain a synchrotron period T s of 100 turns.