piecewise linear time-delay systems.

![\[fig8\] (a) The frequency difference ($\Delta \Omega_{1,i}, 
\quad i=10,20,30,40$ and $50$) and (b) the frequency ratio ($\Omega_i/\Omega_1, 
\quad i=10,20,30,40$ and $50$) of uniformly selected systems are plotted as a function of the coupling strength $\varepsilon \in (0,1.0)$. Each line corresponds to the difference/ratio between the response system and the drive system. The black filled circles indicate the average frequency difference/ratio of all the ($N-1$) response systems from the drive system.](piece_phase_freq_diff_ratio){width="0.6\columnwidth"}

It is also to be noted that the phenomenon remains qualitatively the same even when the total number of oscillators in the array is increased. We also wish to emphasize that the results remain qualitatively unaltered even for different sets of random values for the nonlinear parameters, $\alpha_i$, confirming the robustness of our results.

\[sec:level2c\]Confirmation of GPS from the original untransformed non-phase-coherent attractor
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Next, we use two different approaches, namely recurrence quantification analysis [@romano05; @marwan07] and the concept of localized sets [@pereira07] without estimating the phase explicitly to prove the existence of GPS from the original nonphase coherent hyperchaotic attractors.

### \[sec:level2d\]Recurrence quantification for GPS

Now, we use the recurrence quantification measures discussed in Sec. \[rqa\] in characterizing the existence of GPS via sequential synchronization. 