Parametric optics for FUDU lattices with strongly focusing undulators

In low-energy FEL beamlines, like SXFEL-SBP at the Shanghai Synchrotron Radiation Facility and FLASH1 and FLASH2 at DESY, SASE undulators with perfectly reasonable strength may dynamically affect the optics of the Focusing-Undulator-Defocusing-Undulator (FUDU) cells, pre-matched for a given fixed set of undulator parameters, so violently that a dynamical re-adjustment of the FUDU quadrupoles becomes mandatory. Here we refine and generalize a result reported at the FEL conference 2024. Our almost-analytical result allows implementation in the control system, and is valid for fairly general symplectic coupling-free perturbing matrices. In an approximative sense it includes undulators changing along the beamline and even missing undulators in given cells.