Synchrotron motion with rf phase modulation was studied experimentally. Poincarb maps in the resonant precessing frame were obtained from the experimental data and compared with the tori of the resonant Hamiltonian. Our experimental data revealed island structure in longitudinal phase space. Experimental results for synchrotron motion excited by phase modulation at the third harmonic of the synchrotron frequency are also reported. '

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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.

We investigate effects of quantum fluctuation, potential well distortion, quantum lifetime, and Touschek lifetime of the quasi-isochronous ͑QI͒ dynamical system. The Fokker-Planck equation is employed to study the equilibrium bunch distribution. The quantum lifetime in the moderate damping regime is compared with analytical formulae. The effects of harmonic radio-frequency phase modulation on equilibrium distribution function, quantum lifetime reduction, and the occurrence of stochastic resonance are studied. The formula for the Touschek lifetime for the QI dynamical system is derived and studied.

The synchrotron equation of motion in quasi-isochronous ͑QI͒ storage rings is transformed to a universal Weierstrass equation, where the solution is given by Jacobian elliptic functions. Scaling properties of the QI Hamiltonian are derived. The effects of phase space damping and the sensitivity of particle motion to external harmonic modulation are studied. We find that the rf phase modulation is particularly enhanced in QI storage rings. Exact formula and sum rules for resonance strength coefficients are derived. When the QI dynamical system is subject to harmonic modulation, it exhibits a sequence of period-2 bifurcations leading to global chaos in a region of modulation tune. This means that the operators of QI storage rings should pay special attention to rf phase noise. ͓S1063-651X͑96͒06707-4͔