Nonlinear dynamics issues for quasi-isochronous storage rings

The synchrotron equation of motion in quasi-isochronous (QI) storage rings was transformed to a universal Weierstrass equation, where solution is given by Jacobian elliptic functions. Scaling properties of QI Hamiltonian were derived. The effects of phase space damping and the sensitivity of particle motion to external harmonic modulation were studied. We found that rf phase modulation is particularly enhanced in QI storage rings. This means that the operators of QI storage rings should pay special attention to rf phase modulation. Exact formulae and sum rules for resonance strength coefficients were derived. In the presence of radiation damping and rf phase modulation, the QI system exhibits a sequence of period-two bifurcations enroute to global chaos (instability) in a region of modulation tune. The critical modulation amplitude for the onset of global chaos shows a cusp as a function of modulation tune. This cusp was shown to arise from the transition from the 2:1 to the 1:1 parametric resonances. We also studied the effect of rf voltage modulation and found that the tolerance of rf voltage modulation is much larger than that of rf phase modulation.