New type of bunch lengthening with cusp catastrophe in electron storage rings

Using a simple wake function, it is shown that the equilibrium bunch length in electron storage rings with localized wake can have a cusp-catastrophe behavior. Contrary to the results of the conventional theory for a distributed wake force, the system becomes bistable (period 1 and period 2) and exhibits hysteresis in some region of the parameter space. These features are predicted by a Gaussian approximation and confirmed by multiparticle tracking