We numerically study a directed small-world network consisting of
attractively coupled, identical phase oscillators. While complete
synchronization is always stable, it is not always reachable from random
initial conditions. Depending on the shortcut density and on the asymmetry of
the phase coupling function, there exists a regime of persistent chaotic
dynamics. By increasing the density of shortcuts or decreasing the asymmetry of
the phase coupling function, we observe a discontinuous transition in the
ability of the system to synchronize. Using a control technique, we identify
the bifurcation scenario of the order parameter. We also discuss the relation
between dynamics and topology and remark on the similarity of the
synchronization transition to directed percolation.Comment: This article has been accepted in AIP, Chaos. After it is published,
it will be found at http://chaos.aip.org/, 12 pages, 9 figures, 1 tabl