have introduced a nonlinear transformation to recast the original non-phase-coherent attractor into a smeared limit cycle-like attractor with a single center of rotation. Our transformation is obtained by defining a new state variable [@senthilkumar06; @senthilkumar08], $$z(t+\tau) = x(t)x(t+\hat{\tau})/x(t+\tau),
\label{eq_2}$$ where $\hat{\tau}$ is the optimal value of time-delay to be chosen in order to avoid any additional center of rotation. Then, the projected trajectory in the transformed state space ($x(t+\tau),z(t+\tau)$) will resemble that of a smeared limit cycle-like attractor. Such an approach facilitates the estimation of phase from the conventional techniques.

\[rqa\]Recurrence quantification analysis
-----------------------------------------

Several measures of complexity quantifying the small scale structures in recurrence plots have been proposed and the corresponding description is known as recurrence quantification analysis (RQA) [@marwan07]. In addition to the other advantages of the RQA such as its application to short experimental data, nonstationary and noisy data, we find that it can also be applied directly to highly complicated non-phase-coherent attractors of the time-delay systems in characterizing the synchronization transitions, in particular phase synchronization (PS) transitions. Among the available recurrence quantification measures, we use the Correlation of Probability of Recurrence (CPR) and the generalized autocorrelation function $P(t)$, which are estimated from the original non-phase-coherent hyperchaotic 