different values of the coupling strength $\varepsilon$.](piece_avg_phase_diff){width="0.6\columnwidth"}

In the absence of coupling ($\varepsilon = 0.0$) between the piecewise linear time-delay systems in the array, the individual subsystems evolve independently in an asynchronous fashion which is indeed confirmed from the monotonous increase in the phase differences, $\Delta\phi_{i} = \phi_{1}-\phi_{i}$, between the drive and the uniformly selected response systems ($i = 10,20,30,50$) indicated by asterisk symbols in Figs. \[fig2\]. Phase slips (marked by filled squares and triangles) in Figs. \[fig2\] for intermediate coupling strengths indicate that the coupled systems are in their transition to phase synchronization with the drive. For suitable values of $\varepsilon$, the response systems in the array are entrained to the drive one sequentially, indicated by filled circles, as seen in Figs. \[fig2\]. To be more precise, it is clear from this figure that, the $10$th oscillator is synchronized first at $\varepsilon = 0.45$, while the other systems away from it are still in their transition to phase synchronization. Further increase in $\varepsilon$ to $\varepsilon=0.55$ results in synchronization of the $20$th oscillator with the drive leaving the remaining far away oscillators from it in their transition state. The $30$th and $50$th oscillators are entrained to the drive at $\varepsilon 