Limits to detection of generalized synchronization in delay-coupled chaotic oscillators

We study how reliably generalized synchronization can be detected and characterized from time-series analysis. To that end, we analyze synchronization in a generalized sense of delay-coupled chaotic oscillators in unidirectional ring configurations. The generalized synchronization condition can be verified via the auxiliary system approach; however, in practice, this might not always be possible. Therefore, in this study, widely used indicators to directly quantify generalized and phase synchronization from noise-free time series of two oscillators are employed complementarily to the auxiliary system approach. In our analysis, none of the indices provide the consistent results of the auxiliary system approach. Our findings indicate that it is a major challenge to directly detect synchronization in a generalized sense between two oscillators that are connected via a chain of other oscillators, even if the oscillators are identical. This has major consequences for the interpretation of the dynamics of coupled systems and applications thereof.

Figure 1

A schematic of the network configuration considered in this study. Mackey-Glass oscillators are chosen as network nodes and arranged in a ring. The oscillators in the ring are delay coupled, with a total delay in the loop of ${\tau }_d$. We focus on elements X, Y, and Z. The part surrounded by gray dashed line is an optional chain of coupled oscillators considered in Refs. [17, 18] and equivalent to oscillators 1 and $n/2$ on the ring configuration.

Figure 2

The time traces of Mackey-Glass oscillators X and Y in the ring configuration network as depicted in Fig. 1. Each panel shows the cases of $n=2,6,8$, and 18.

Figure 3

Power spectra obtained from the time series in Fig. 2. All the panels are plotted in log-log scale.

Figure 4

The cross correlation functions between the time series of X and Y in Fig. 2.

Figure 5

The cross redundancy based on correlation integrals as a function of time shift. From top to bottom, $n=2,6,8$, and 18. We used $5\%$ of a maximum distance on the multidimensional state as a threshold $\varepsilon$ in Eq. (4).

Figure 6

Maximum value of the cross redundancy based on correlation integrals as a function of the number of oscillators in the system. Five percent of a maximum distance on the multidimensional state is adopted as a threshold in all cases.

Figure 7

The coefficient $S$. From top to bottom, $n=2,6,8$, and 18. The parameter $R_R$ is set to $0.1$.

Figure 8

The joint probability of recurrences as a function of the number of oscillators in the system. $R_R=0.1$ is used for all cases.

Figure 9

The generalized synchronization index from the reconstructed state space as a function of time shift. From top to bottom, $n=2,6,8$, and 18. The parameters $R$, $m$, and ${\tau }_e$ are set to 8, 10, and 5, respectively.

Figure 10

Maximum value of the generalized synchronization index as a function of the number of oscillators in the system. In all cases, the parameters $R$, $m$, and ${\tau }_e$ are set to 8, 10, and 5, respectively.

Figure 11

The cross correlation coefficient between the original time series and the predicted time series by the convergent cross mapping as a function of time shift. From top to bottom, $n=2,6,8$, and 18. The parameters $m$ and ${\tau }_e$ are set to 10 and 5, respectively.

Figure 12

Maximum value of the cross correlation coefficient between the original time series and the predicted time series by the convergent cross mapping as a function of the number of oscillators in the system. In all cases, the parameters $m$ and ${\tau }_e$ are set to 10 and 5, respectively.

Figure 13

The phase synchronization in a ring configuration. All indices to quantify phase synchronization as a function of the number of oscillators in the system. For $I({\varphi }_x;\Delta {\varphi }_y|{\varphi }_y)$ and $I({\varphi }_y;\Delta {\varphi }_x|{\varphi }_x)$, the average values of the time lags from ${\tau }_s=1$ to 2000 are plotted. For the CPR, ${\tau }_s=600$ is used and the threshold $\varepsilon$ is selected as $10\%$ of a maximum distance of all pairs of points on the multidimensional state. For the MPC and the CPR indices, both the original values and that for the exchanged data (subscript $ex$) are included.

Figure 14

Histograms of phase differences $\Phi \left(k\right)$ between X and Y estimated by the Hilbert transform. The cases of $n=2,6,8$, and 18 are plotted in each panel.