dynamics. Broad distribution of $\langle \phi_i \rangle$ and $\langle \Omega_i \rangle$ for small $\varepsilon$ correspond to asynchronous states while for $\varepsilon\ge 0.75$ both the average phase and average frequency converge to a single value indicating the existence of GPS. However, for the intermediate values of $\varepsilon$, the main cluster and other small clusters are hardly visible in Figs. \[fig6\]. Nevertheless, these clusters are clearly seen in Figs. \[fig3\] and \[fig5\], where the snapshots of the dynamical organization of all the $N$ oscillators in the array are depicted for specific values of the coupling strength.

![\[fig6\] (a) Time averaged phase ($\langle \phi_i \rangle$) and (b) Time averaged frequency ($\langle \Omega_i \rangle$), $i = 1,2,\cdots, 50$ as a function of the coupling strength $\varepsilon \in (0,1.0)$. Here for each value of $\varepsilon$ we have plotted the average phase/frequency of all the $N$ = 50 oscillators which is shown by the filled circles.](piece_avg_phase_freq_eps){width="0.6\columnwidth"}

The frequency difference ($\Delta \Omega_{1,i}=\Omega_1-\Omega_i$) and the frequency ratio ($\Omega_i/\Omega_1$) for the oscillators with the index $i=10,20,30,40$ and $50$ along with their average (represented by black filled circles) of all the $(N-1)$ response oscillators as a function of the $\varepsilon$ are depicted in Fig. \[fig8\](a) and Fig. \[fig8\](b), respectively. It 