  The time evolution of periodically driven non-Hermitian systems is in general
non-unitary but can be stable. It is hence of considerable interest to examine
the adiabatic following dynamics in periodically driven non-Hermitian systems.
We show in this work the possibility of piecewise adiabatic following
interrupted by hopping between instantaneous system eigenstates. This
phenomenon is first observed in a computational model and then theoretically
explained, using an exactly solvable model, in terms of the Stokes phenomenon.
In the latter case, the piecewise adiabatic following is shown to be a genuine
critical behavior and the precise phase boundary in the parameter space is
located. Interestingly, the critical boundary for piecewise adiabatic following
is found to be unrelated to the domain for exceptional points. To characterize
the adiabatic following dynamics, we also advocate a simple definition of the
Aharonov-Anandan (AA) phase for non-unitary cyclic dynamics, which always
yields real AA phases. In the slow driving limit, the AA phase reduces to the
Berry phase if adiabatic following persists throughout the driving without
hopping, but oscillates violently and does not approach any limit in cases of
piecewise adiabatic following. This work exposes the rich features of
non-unitary dynamics in cases of slow cycling and should stimulate future
applications of non-unitary dynamics.
