  We propose Moebius maps as a tool to model synchronization phenomena in
coupled phase oscillators. Not only does the map provide fast computation of
phase synchronization, it also reflects the underlying group structure of the
sinusoidally coupled continuous phase dynamics. We study map versions of
various known continuous-time collective dynamics, such as the synchronization
transition in the Kuramoto-Sakaguchi model of non-identical oscillators,
chimeras in two coupled populations of identical phase oscillators, and
Kuramoto-Battogtokh chimeras on a ring, and demonstrate similarities and
differences between the iterated map models and their known continuous-time
counterparts.
