to be 1.6 [@senthilkumar06]. The same transformation has also been performed for the non-phase-coherent chaotic attractor of the Mackey-Glass time-delay system with an appropriate $\hat{\tau}$ in Ref. [@suresh10]. The hyperchaotic nature of the attractor in Figs. \[fig1\] for the above parameter values is confirmed from three positive Lyapunov exponents at $\tau = 15.0$ in the spectrum of ten maximal Lyapunov exponents of (\[eq\_1\]a) as a function of the time-delay parameter $\tau \in (3, 25)$ in Fig. \[fig1b\].

![image](piece_phase_diff){width="1.0\columnwidth"}

\[sec:level2b\]Identification of GPS from the transformed attractor of the piecewise linear time-delay system
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To confirm the existence of GPS in the array of piecewise linear time-delay systems, we have fixed size of the array as $N=50$. The transformed smeared limit cycle-like attractors with a single center of rotation of some of the uniformly selected subsystems ($i$ = 10, 20, 30, 50) in the linear array are shown in Figs. \[fig1\] (c-f). We have used the conventional Poincaré section technique [@pikovsky01; @boccaletti02] to estimate the instantaneous phases of all the oscillators after the transformation of the attractors, which now yields monotonically increasing phase. Open circles in Figs. \[fig1\](b-f) indicate the Poincaré section.

![\[fig2a\] The evolution of the average phase difference \[$\eta(t)$\] for 