Effect of magnetized electron cooling on a Hopf bifurcation

We have observed longitudinal limit cycle oscillations of a proton beam when a critical threshold in the relative velocity between the proton beam and the cooling electrons has been exceeded. The threshold for the bifurcation of a fixed point into a limit cycle, also known as a Hopf bifurcation, was found to be asymmetric with respect to the relative velocity. Further experiments were performed to verify that the asymmetry was related to electron beam alignment with respect to the stored proton beam. The measured amplitudes of the ensuing limit cycle were used to determine the cooling drag force, which exhibits the essential characteristics of the magnetized cooling, where the limit cycle attractor can coexist with a damping-free region and/or a fixed point attractor. {copyright} {ital 1996 The American Physical Society.}