Prediction of long-term beam stability using high order toroidal invariants

It is noted that two ways have been proposed for detecting signs of long-term beam instability. They both rest on the belief that if a particle is long-term unstable, it will experience a significant and discernible amount of chaotic motion. This chaotic motion can be detected by (a) looking for small departure from regular motion such as slow change of an action invariant or (b) tracking two neighboring particle and looking for an exponential growth of their phase space separation. The authors describe efforts of type (a) and present some recent comparisons with procedures of type (b). Results are presented for the Superconducting Super Collider