Halo formation for a test particle in a mismatched KV beam is studied. Parametric resonances of the particle Hamiltonian due to envelope modulation are studied with particular emphasis on period 2 resonance which plays dominant role in Halo formation. It is shown that the onset of global chaos exhibits a sharp transition when the amplitude of modulation is larger than a critical value which is a function of a single parameter, , i.e., the ratio of the space charge perveance to the focusing strength.

Controlled beam-bunch dilution has been proved to be essential in the acceleration of high intensity beams in many high energy accelerators such as the AGS and the CERN PS for transition energy crossing. A secondary RF cavity operating at 200 MHz has been used routinely to blow up the longitudinal phase space for the transition energy crossing. In our recent experiments, we have observed a non-diffusive mechanism in the beam dilution process. As part of a new proposal, we will devote more effort to understand the importance of parametric resonances in bunched-beam dilution.Double RF systems were used in accelerator physics to decrease space-charge effects by reducing the peak current or to overcome multi-bunch instability by modifying the time structure of the beam. Double RF systems were also found to be a very effective tool for increasing synchrotron tune spread and enchancing the Landau damping rate. In a series of Cooler experiments, we investigated a possibile use of double RF systems in order to stimulate longitudinal beam dilution by modulating the secondary high frequency RF cavity. The experiment described in this paper was motivated by recent theoretical and experimental studies in the field.This progress report discusses experimental measurements of the evolution of the bunch profile while a modulating secondary RF system is acting on the beam. The experiment was conducted at IUCF and used a bunched proton beam with a kinetic energy of 45 MeV. The revolution frequency for the synchronous particle was fo = 1.03168 MHz.The frequency of the primary RF cavity was 1.03168 MHz with hl = 1, and the harmonic number of the secondary rf cavity was h = 9. The voltage of the main RF cavity was set at about 300 V, which resulted in a synchrotron frequency of about 705 Hz while operating with the primary RF cavity alone. The total beam current was about 100 PA, or equivalently 6 x 10' protons per bunch. The accelerator was operated with a cycle time of 10 s; the injected beam was electron-cooled for about 3 s (the rms equilibrium bunch length was about 10 -20 ns); then the phase modulation was applied and beam profiles were digitized for a few hundred milliseconds. To measure the time evolution of the bunch, we used a BPM sum signal passing through a low loss cable (7/B1' heliax high bandwidth, low attenuation cable). Then the signal was amplified by an amplifier with a bandwidth of 150 MHz and digitized by a fast sampling scope at a time step of 1 ns. The system

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