The Hopf's limiting cycle -- A method to measure the electron cooling force

The longitudinal phase space structure of protons in the presence of an electron drag force is studied. Depending on the relative difference in average proton velocity and electron velocity, the fixed point near the center of the phase space can be stable or unstable. When it is unstable, a stable Hopf's limit cycle exists. By measuring the appearance and size of the Hopf's limit cycle, the electron cooling force can be deduced