cluster states for different values of coupling strength. The different shapes (colors) indicate that the corresponding nodes are phase synchronized. (a) non-phase synchronized case for $\varepsilon$ = 0.0 (b) First fourteen oscillators in Eq. (\[eq\_1\]b) are phase synchronized with the drive system for $\varepsilon$ = 0.45. (c), (d) and (e) Sequential phase synchronization and the formation of small cluster states for $\varepsilon$ = 0.51, 0.58 and 0.62, respectively, and (f) global phase synchronization for $\varepsilon$ = 0.8.](piece_node_node_50){width="0.8\columnwidth"}

The nature of dynamical organization of local oscillators to form GPS can be better understood by examining the time average phase and the time average frequency as a function of the oscillator index for different values of the coupling strength. For this purpose, we define the time averaged phase ($\langle \phi_i \rangle$) as $$\big\langle \phi_{i}(t)\big\rangle=\Big\langle2\pi k+2\pi \frac{t^{i}-t^{i}_{k}}{t^{i}_{k+1}-t^{i}_{k}}\Big\rangle_{t},~(t^{i}_{k}<t^{i}<t^{i}_{k+1}),
\label{eq3a}$$ and the time averaged frequency as ($\langle \Omega_i \rangle$) $$\big\langle \Omega_{i}(t)\big\rangle = \lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} \dot\phi_{i}(t) dt,
\label{eq3a1}$$ where $t_{k}$ is the time of the $k^{th}$ crossing of the flow with the Poincaré section of the $i^{th}$ attractor and $\langle...\rangle_{t}$ denotes time average.

Figures \[fig3\](a) and \[fig3\](b) depict the average phase and the average frequency, respectively, for different $\varepsilon$ as a function of 