Bifurcation of resonance islands and Landau damping in the double-rf system

The attractors of a double-rf system subject to rf phase modulation, in the presence of a weak damping force, were measured as a function of the modulation frequency. We found that the phase amplitudes of the attractors followed a simple predictable path related to the synchrotron tune of the double-rf system. These attractors were found to bifurcate at a modulation frequency near the maximum synchrotron frequency. We also found that the coherent synchrotron oscillations decohered rapidly at small synchrotron amplitudes but showed little decoherence at large synchrotron amplitudes. The experimental result has some implications for the Landau damping of coherent beam instabilities