Maps for coupled electron and ion clouds in accelerators

Electron clouds sometimes limit accelerator performance by their appearance when the circulating bunch population exceeds a threshold value. Dynamical models of cloud buildup, and of the phase transition from ``cloud off'' to ``cloud on'', are enhanced when simple coupling between electron and ion clouds is included. Maps are then capable of reproducing the first order phase transitions sometimes seen in practice. They also predict that hysteresis, period doubling, and chaotic phenomena may be observed.