  The notion of phase synchronization in time-delay systems, exhibiting highly
non-phase-coherent attractors, has not been realized yet even though it has
been well studied in chaotic dynamical systems without delay. We report the
identification of phase synchronization in coupled nonidentical piece-wise
linear and in coupled Mackey-Glass time-delay systems with highly
non-phase-coherent regimes. We show that there is a transition from
non-synchronized behavior to phase and then to generalized synchronization as a
function of coupling strength. We have introduced a transformation to capture
the phase of the non-phase coherent attractors, which works equally well for
both the time-delay systems. The instantaneous phases of the above coupled
systems calculated from the transformed attractors satisfy both the phase and
mean frequency locking conditions. These transitions are also characterized in
terms of recurrence based indices, namely generalized autocorrelation function
$P(t)$, correlation of probability of recurrence (CPR), joint probability of
recurrence (JPR) and similarity of probability of recurrence (SPR). We have
quantified the different synchronization regimes in terms of these indices. The
existence of phase synchronization is also characterized by typical transitions
in the Lyapunov exponents of the coupled time-delay systems.
