  We numerically study the celebrated Kuramoto model of identical oscillators
arranged on the sites of a two-dimensional periodic square lattice and subject
to nearest neighbor interactions and dichotomous noise. In the nonequilibrium
stationary state attained at long time, the model exhibits a
Berezinskii-Kosterlitz-Thouless ($BKT$)-like transition between a phase at low
noise amplitude characterized by quasi long-range order (critically ordered
phase) and algebraic decay of correlations and a phase at high noise amplitude
that is characterized by complete disorder and exponential decay of
correlations. The interplay between the noise amplitude and the noise
correlation time is investigated, and the complete, nonequilibrium
stationary-state phase diagram of the model is obtained. We further study the
dynamics of a single topological defect for various amplitude and correlation
time of the noise. Our analysis reveals that a finite correlation time promotes
vortex excitations, thereby lowering the critical noise amplitude of the
transition with an increase in correlation time. In the suitable limit, the
resulting phase diagram allows to estimate the critical temperature of the
equilibrium $BKT$ transition, which is consistent with that obtained from the
study of the dynamics in the Gaussian white noise limit.
