  Coupled oscillators with time-delayed network interactions are critical to
understand synchronization phenomena in many physical systems. Phase reductions
to finite-dimensional phase oscillator networks allow for their explicit
analysis. However, first-order phase reductions - where delays correspond to
phase lags - fail to capture the delay-dependence of synchronization. We
develop a systematic approach to derive phase reductions for delay-coupled
oscillators to arbitrary order. Already the second-order reduction can predict
delay-dependent (bi-)stability of synchronized states as demonstrated for
Stuart-Landau oscillators.
