boundary conditions. In Secs. \[sec:level4\] and \[sec:level5\] we consider the linear array of two piecewise linear time-delay systems with closed end boundary conditions (ring topology) and discuss the occurrence of partial phase synchronization and finally we summarize our results in Sec. \[sec:level6\].

![\[fig1\] Attractors of the piecewise linear system. (a) non-phase-coherent hyperchaotic attractor of the drive system (\[eq\_1\]a). (b) Transformed attractor of the drive system. (c-f) Transformed attractors of some of the uniformly selected response systems ($i = 10,20,30,50$) in the new state space ($x_i(t+\tau),z_i(t+\tau)$).](piece_phase_att){width="0.5\columnwidth"}

\[measures\]Estimates of phase
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Identifying Chaotic phase synchronization is a nontrivial problem in time-delay systems exhibiting complicated hyperchaotic attractors. For this purpose one requires appropriate measures. In this section, we describe briefly the various measures which we used from three different frameworks in this paper to estimate the phase of time-delay systems thereby facilitating the characterization of phase synchronization between the coupled time-delay systems.

Transformation of the original non-phase-coherent attractor
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Time-delay systems usually exhibit a highly complicated non-phase-coherent chaotic/hyperchaotic attractor with their flows having more than a single center of rotation. Such an attractor does not yield a monotonically increasing phase on estimating it using the conventional techniques [@senthilkumar06; @senthilkumar08]. For this purpose, we 