is also clearly seen from these figures that the nearby oscillators are synchronized sequentially as $\varepsilon$ is increased, that is, $i=10$ is synchronized first and then $i=20$, and so on as $\varepsilon$ is increased. Finally, all the $N=50$ oscillators in the array are phase synchronized for $\varepsilon\ge 0.75$ attributing to the emergence of GPS via local sequential synchronization.

The well-known Kuramoto order parameter [@moreno04] can also be used for quantifying the global phase synchronization in the array, which is defined as $$R~e^{i\psi} = \Big\langle\Big|\frac{1}{N}\sum_{j=1}^{N} e^{i\phi_{j}(t)}\Big|\Big\rangle_t.
\label{eq_kop}$$ Here $\phi_{j}(t)$ represents the instantaneous phase of the $j^{th}$ system, $\psi(t)$ is the average phase and $\langle...\rangle_t$ denotes a time average. When all the systems are in a phase synchronized state, the value of $R\approx 1.0$. Figure \[fig10\] shows that $R$ decreases at first for small increase in $\varepsilon$ as observed in Figs. \[fig8\](a) and \[fig8\](b) accounting for the increased randomness in the phases of the coupled systems as seen in the average phase and the average frequency of all the oscillators in Figs. \[fig6\]. Further increase in $\varepsilon$ results in a gradual increase in the value of $R$, reaching $R\approx 1.0$ for $\varepsilon>0.75$ confirming the existence of GPS in the array of coupled 