Effects of overlapping parametric resonances on the particle diffusion process

Quadrupole-mode transfer function and the nonlinear Mathieu instability

We show that the quadrupole-mode transfer function (QTF) is a powerful nondestructive tool to measure properties of dynamical systems. In particular, we discuss the feasibility of using the QTF to measure the betatron tunes and the beam emittances with a beam-position monitor system. The QTF can also be used to compensate the optical mismatch during the beam injection process. However, it is less effective than the rf dipole method in overcoming the intrinsic spin resonances for polarized beam acceleration.