($i=10,20,30$ and $50$) response systems with the drive system. The filled circles correspond to the mean value of the CPR of all the ($N-1$) piecewise linear systems in the array.](piece_cpr_50){width="0.7\columnwidth"}

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

### \[sec:level2e\]GPS using the concept of localized sets

Next, we use the framework of localized sets described in Sec. \[ls\] to demonstrate the existence of GPS via sequential phase synchronization. We defined the event as a Poincaré section chosen at $x(t-\tau)=0.8$ and $x(t)<0.8$ on the attractor. The sets (indicated as filled circles) obtained by observing the drive system ($i = 1$) whenever the defined event occurs in the response system $i = 10$ along with the drive attractor are depicted in the first row in Fig. \[fig13\]. Similarly, the sets obtained by observing the response systems ($i = 10, 30$ and $50$) whenever the event occurs in the drive system are depicted along with the corresponding response attractors in the other subsequent rows.

The sets spread over the entire attractors in Figs. \[fig13\](a-d) for $\varepsilon = 0.0$ confirm the asynchronous evolution of the subsystems in the array. The sets localized in a large area of the attractors in Figs. \[fig13\](e-f) for $\varepsilon = 0.4$ indicate the transition to CPS 