  The appearance of so-called exceptional points in the complex spectra of
non-Hermitian systems is often associated with phenomena that contradict our
physical intuition. One example of particular interest is the state-exchange
process predicted for an adiabatic encircling of an exceptional point. In this
work we analyse this and related processes for the generic system of two
coupled oscillator modes with loss or gain. We identify a characteristic system
evolution consisting of periods of quasi-stationarity interrupted by abrupt
non-adiabatic transitions, and we present a qualitative and quantitative
description of this switching behaviour by connecting the problem to the
phenomenon of stability loss delay. This approach makes accurate predictions
for the breakdown of the adiabatic theorem as well as the occurrence of chiral
behavior observed previously in this context, and provides a general framework
to model and understand quasi-adiabatic dynamical effects in non-Hermitian
systems.
