  We consider a long-range model of coupled phase-only oscillators subject to a
local potential and evolving in presence of thermal noise. The model is a
non-trivial generalization of the celebrated Kuramoto model of collective
synchronization. We demonstrate by exact results and numerics a surprisingly
rich long-time behavior, in which the system settles into either a stationary
state that could be in or out of equilibrium and supports either global
synchrony or absence of it, or, in a time-periodic synchronized state. The
system shows both continuous and discontinuous phase transitions, as well as an
interesting reentrant transition in which the system successively loses and
gains synchrony on steady increase of the relevant tuning parameter.
