Figure \[fig11\] depicts the generalized autocorrelation function of the drive $P_1(t)$ and that of some of the response systems $P_{10}(t)$, $P_{30}(t)$, and $P_{50}(t)$ for different $\varepsilon$. For $\varepsilon=0.0$, none of the maxima of $P_i(t)$ coincide in Fig. \[fig11\](a) indicating asynchronous state. $P_i(t)$ for $i=1,10,30$ and $50$ are plotted in Figs. \[fig11\](b) and  \[fig11\](c) for $\varepsilon=0.45$ and $\varepsilon=0.8$, respectively. In Fig. \[fig11\](b), the maxima of $P_1(t)$ and $P_{10}(t)$ are in perfect agreement (see Fig. \[fig11\](b)(i)) confirming the existence of phase synchronization

![\[fig10\] Plot of the phase order parameter ($R$) as a function of the coupling strength indicating global phase synchronization in the array of ($N$ = 50) coupled piecewise linear time-delay systems (\[eq\_1\]).](piece_op){width="0.7\columnwidth"}

between the oscillators with the index $i=1$ and $i=10$, which is in agreement with the results of the earlier section. On the other hand, some of the maxima of $P_1(t)$ and $P_{30}(t)$ in Fig. \[fig11\](b)(ii) are in agreement attributing to the transition to CPS among them and none of the maxima of $P_1(t)$ and $P_{50}(t)$ in Fig. \[fig11\](b)(ii) are in agreement corresponding to independent evolution, which confirms the sequential synchronization. All the maxima of $P_i(t)$ are in complete agreement in Fig. \[fig11\](c) confirming the existence of GPS. Also, 