Alpha Buckets in Longitudinal Phase Space: a Bifurcation Analysis

At HZB’s BESSY II and PTB’s Metrology Light Source (MLS) facilities we have the ability to tune the momentum compaction factor $\alpha$ up to second non-linear order. The non-linear dependence $\alpha (\delta )$ brings qualitative changes to the longitudinal phase space and introduces new fix points $\alpha (\delta )=0$ which produce the so-called $\alpha$-buckets. We present with this paper an analysis of this phenomena from the standpoint of bifurcation theory. With this approach we were able to characterize the nature of the fix points and their position in direct dependence on the tunable parameters. Furthermore, we are able to place stringent conditions onto the tunable parameters to either create or destroy $\alpha$-buckets.