= 0.62$ and $0.79$, respectively, illustrating the existence of GPS through the sequential synchronization of the subsystems locally in the array.

![\[fig3\] (a) Time averaged phase $\langle \phi_i \rangle$ given by (\[eq3a\]) and (b) time averaged frequency $\langle \Omega_i \rangle$ given by (\[eq3a1\]) of all the systems in the array (\[eq\_1\]) with open end boundary conditions plotted as a function of the system index $i$ for different values of the coupling strength $\varepsilon$.](piece_avg_phase_freq){width="0.7\columnwidth"}

The existence of GPS is also confirmed by the average phase difference, $\eta(t)$, defined as $$\eta(t) = \frac{1}{N-1} \sum_{j=2}^{N} (\phi_{1}-\phi_{j}).
\label{eq_3}$$ A global measure to characterize the existence of GPS is the average phase difference ($\eta(t)$), which is shown in Fig. \[fig2a\] for different values of $\varepsilon$. The monotonous increase in $\eta(t)$ for $\varepsilon=0.0$, correspond to the independent evolution of all the subsystems in the array. Phases of the nearby oscillators are mutually locked sequentially as $\varepsilon$ is gradually increased, which is indicated by the low degrees of $\eta(t)$ for $\varepsilon=0.56$ and $0.77$ in Fig. \[fig2a\]. Finally, $\eta(t)\approx 0$ at $\varepsilon=0.79$ corroborating the existence of GPS in the array.

![\[fig5\] Oscillator index vs oscillator index snap shots indicating the sequential phase synchronization and the dynamical organization of 