Parametric resonances in synchrotrons with two rf systems.

When a double rf system is subjected to sinusoidal phase modulation, the Poincare surfaces of the section display a rich spectrum of resonance islands. Stable and unstable fixed points of these resonance islands form a tree of bifurcation branches which can be explained as parametric resonances generated by external phase modulation. A semianalytic determination of the condition for the bifurcation of fixed points is presented for an autonomous Hamiltonian of one degree of freedom with sinusoidal time dependent perturbation.