  Phase-locked states with a constant phase shift between the neighboring
oscillators are studied in rings of identical Kuramoto oscillators with
time-delayed nearest-neighbor coupling. The linear stability of these states is
derived and it is found that the stability maps for the dimensionless equations
show a high level of symmetry. The size of the attraction basins is numerically
investigated. These sizes are changing periodically over several orders of
magnitude as the parameters of the model are varied. Simple heuristic arguments
are formulated to understand the changes in the attraction basin sizes and to
predict the most probable states when the system is randomly initialized.
