Nonlinear instabilities in multiple time stepping

We have recently analyzed Verlet-I/r-RESPA and discovered that there is a nonlinear resonance [47,60] which in practice means that the time step possible for the method is more restricted than the linear analysis suggests. The method becomes unstable at roughly a third of the fastest period, rather than half as the linear analysis suggests. For the same case of flexible water that means there is instability at around 3fs. This is not much better than leapfrog.