Effects of overlapping parametric resonances on the particle diffusion process

The evolution of the beam distribution in a double-rf system with a phase m odulation on either the primary or secondary rf cavity was measured. We fin d that the particle diffusion process obeys the Einstein relation if the ph ase space becomes globally chaotic. When dominant parametric resonances sti ll exist in the phase space, particles stream along the separatrices of the dominant resonance, and the beam width exhibits characteristic oscillatory structure. The particle-tracking simulations for the double-rf system are employed to reveal the essential diffusion mechanism. Coherent octupolar mo tion has been observed in the bunch beam excitation. The evolution of the l ongitudinal phase space in the octupole mode is displayed. [S1063-651X(99)0 0111-7].