Figure 1

Estimation of the delay in coupling of noisy limit cycle oscillators [Eqs. (18)]. The dotted vertical lines mark the true value of the delay $T$. Left panel: error of the fit $\mathcal{E}$ as the function of shift $\tau$, computed from the phases obtained via the Hilbert transform (solid line) and from the phases obtained directly from Eqs. (18) (dotted line), as well as the theoretical curve according to Eq. (17) (bold line). Right panel: dependence of the synchronization index $\rho$ on the shift $\tau$ (bold line) and the cross-correlation function (solid line); $\rho \left(\tau \right)$ nearly coincides with the envelope of $C\left(\tau \right)$. Note that $\rho \left(\tau \right)$ dependence provides a biased estimate of the delay.Reuse & Permissions

Figure 2

Unidirectional delayed coupling of chaotic Rössler oscillators [Eqs. (19)]. The delay in coupling is estimated from the position of the minimum of $\mathcal{E}\left(\tau \right)$ and maximum of $\rho \left(\tau \right)$. The dashed curve in the right panel shows the envelope of the cross-correlation function between $x_1$ and $x_2$.Reuse & Permissions

Figure 3

(Color online) Unidirectional delayed coupling of noisy limit cycle oscillators [Eqs. (18)]. The value of the delay in coupling is revealed from the dependences of the fitting error ${\mathcal{E}}_N$ and synchronization index $\rho$ versus shift $\tau$ between the two time series of phases (left and right panels, respectively). The dotted vertical lines mark the true value of the delay $T$. Bold (blue) line illustrates the case of the noisy driver and noise-free driven system; dashed (red) line corresponds to the case when the response system is noisy as well. Dot-dashed line (black) shows the results for the case when both systems are noisy and the observed signals are contaminated by observational noises. Finally, the dotted (black) line shows the results for the latter case obtained after smoothing the signals as described in the text; the fact that the curves obtained after smoothing nearly coincide with the curves for noise-free response system demonstrates that the delay can be efficiently estimated from real-world data. (In the right panel three of four curves nearly coincide.) Note that $\rho \left(\tau \right)$ dependence provides a biased estimate of the delay.Reuse & Permissions

Figure 4

(Color online) The influence of the driving strength on the estimation of delay by means of the MFE (left panel) and SIE (right panel) algorithms for unidirectionally coupled noisy maps [Eq. (20)]. Gray scales (colors) code the values of the fitting error $\mathcal{E}\left(\tau \right)$ and synchronization index $\rho \left(\tau \right)$, respectively. Note that the MFE algorithm provides accurate estimates of the delay as long as the driving strength $\epsilon$ exceeds the noise level in the driven system [cf. Eq. (16)]. The SIE algorithm produces a biased estimate, in accordance with Eq. (13).Reuse & Permissions

Figure 5

(Color online) The influence of the noise intensity in the driver on the estimation of delay by means of the MFE (left panel) and SIE (right panel) algorithms for unidirectionally coupled noisy maps [Eq. (20)]. Note that the noise facilitates the identification of delay.Reuse & Permissions

Figure 6

(Color online) The influence of detuning on the estimation of delay by means of the MFE (left panel) and SIE (right panel) algorithms for unidirectionally coupled noisy maps [Eq. (20)]. Note that with an increase of the detuning, the SIE algorithm fails, while the MFE algorithm provides the correct estimate in the whole range of the detuning.Reuse & Permissions

Figure 7

Bidirectional delayed coupling of Van der Pol oscillators [Eqs. (19)]. The dotted vertical lines mark the true value of the delays $T_{12}$ and $T_{21}$.Reuse & Permissions

Figure 8

Schematic illustration of the correlation between time series ${\varphi }_{1,2}$. Phase ${\varphi }_1\left(t_0\right)$ is tightly correlated with ${\varphi }_2(t_0-T_{12})$. Therefore, quantification of the interrelation between the shifted phases provides an extremum for shift $\tau =T_{12}$. On the other hand, ${\varphi }_1\left(t_0\right)$ is tightly correlated with ${\varphi }_2(t_0-2T_{12}-T_{21})$, and hence, the shift $\tau =T_{12}+(T_{12}+T_{21})$ also provides an extremum.Reuse & Permissions

Figure 9

Unidirectional driving with multiple delays [Eqs. (22)]. The dashed line in the right panels shows the envelope of the cross-correlation function. Note that the MFE approach correctly indicates the number of delayed interactions.Reuse & Permissions

Figure 10

The theoretical dependence of the error of the optimal fit on the shift between the time series of phases [see Eq. (B2)]. The position of the minimum of the error coincides with the true value of the delay $T=50$. The value of the minimum is determined by the noise level in the driven system.Reuse & Permissions

Figure 11

(Color online) Two-dimensional discrete Fourier transform $F^2(k_1,k_2)$ of the phase increment $\Delta {\varphi }_1=\Delta {\varphi }_1({\varphi }_1,{\varphi }_2)$ for two unidirectionally coupled Stuart-Landau (left panel) or Rössler (right panel) oscillators [see Eqs. (18, 19)]. Gray scales (colors) code the magnitude of the (significant) Fourier coefficients. Only the first ten harmonics are shown.Reuse & Permissions