Consistency properties of chaotic systems driven by time-delayed feedback

Consistency refers to the property of an externally driven dynamical system to respond in similar ways to similar inputs. In a delay system, the delayed feedback can be considered as an external drive to the undelayed subsystem. We analyze the degree of consistency in a generic chaotic system with delayed feedback by means of the auxiliary system approach. In this scheme an identical copy of the nonlinear node is driven by exactly the same signal as the original, allowing us to verify complete consistency via complete synchronization. In the past, the phenomenon of synchronization in delay-coupled chaotic systems has been widely studied using correlation functions. Here, we analytically derive relationships between characteristic signatures of the correlation functions in such systems and unequivocally relate them to the degree of consistency. The analytical framework is illustrated and supported by numerical calculations of the logistic map with delayed feedback for different replica configurations. We further apply the formalism to time series from an experiment based on a semiconductor laser with a double fiber-optical feedback loop. The experiment constitutes a high-quality replica scheme for studying consistency of the delay-driven laser and confirms the general theoretical results.

Figure 1

Correlation functions from time series of delay-coupled logistic maps, Eq. (19) for $k=0.3$ and $\tau =100$, showing characteristic peak structures. (a) Autocorrelation function of closed-loop system. (b) Cross-correlation function between closed-loop and open-loop system. (c) auto-correlation function of open-loop system. (d)–(f) Zoom into the peak structures from (a)–(c), respectively. In this example, delay time and peak position coincide.

Figure 2

Basic structure for the analytics. Left: Coupling scheme corresponding to the auxiliary system approach. Right: Symbolic representation of input and output for time series from $X$ and $Y$.

Figure 3

Illustration of an ensemble of $J$ replica systems attached to a single closed-loop unit.

Figure 4

Graphical illustration of the relationships between $\alpha$ and $\beta$ given a fixed value of ${\gamma }^2=0.5$ (top), ${\gamma }^2=0.95$ (bottom left), and ${\gamma }^2=0.1$ (bottom right). Black solid line: identity $\alpha =\beta$. Red (gray) solid line: neutral line ${\beta }_0=\alpha {\gamma }^2$. Shaded region: generally allowed relationships according to consistency theory. The intersections of the ellipse with the linear functions are at its outmost extensions with tangents $\alpha =\gamma$ and $\beta ={\gamma }^2$ (black dashed lines in top panel).

Figure 5

Correlations of logistic map with feedback Eq. (19); see text. $\alpha$ and $\beta$ are the values of the autocorrelation functions of $x$ and $y$, respectively, taken at time shift $\mathrm{\Delta }t=\tau =30$. The consistency correlation ${\gamma }^2$ and its root are plotted to validate the outer extensions of the ellipse on the $\alpha$- and $\beta$-axis. The additional ${\gamma }_e$ (circles) is the correlation between $x$ and the ensemble mean of replicas [using Eqs. (20, 21) with $J=1000$ and Eq. (18)], which coincides with $\sqrt{{\gamma }^2}$ from a single replica.

Figure 6

Examples of correlation parameters from the coupled logistic maps, see Eq. (19), drawn in the corresponding ellipse. Left: entire cross-correlation function between $x_t$ and $y_t$, including the first delay echoes around the delay time $\tau =30$. The value at zero time shift corresponds to ${\gamma }^2$ and the vertical extent of the ellipse, $\alpha$ ($\beta$) is picked up from the peaks at the negative (positive) time shift. Right: Corresponding ellipses (yellow/light gray) with entire delay echoes plotted inside (red/dark gray) including the neutral line ${\beta }_0=\alpha {\gamma }^2$ (gray, dashed).

Figure 7

Pairwise projections of the consistent component $f$ and the inconsistent components $n_x,n_y$ of the responses $x$ and $y$, obtained from Eq. (20) for delayed logistic maps with delay $\tau =30$. Time shifts $\mathrm{\Delta }t$ range from $-40$ to 40 to include $\tau$. (a) Projection of the consistent components. (b) Consistent component $f$ and inconsistent component $n_x$ showing the peak related to the closed-loop connection. (c) No correlation between $f$ and $n_y$ in open loop. (d) No correlation between inconsistent responses $n_x$ and $n_y$ due to independence. (e, f) Autoprojection of $n_x$ and $n_y$, respectively, identical and without delay echoes.

Figure 8

Simplified scheme of the double-delay-loop experiment. The laser is subject to optical feedback from either a short delay line or a long one. A pulse generator in combination with two Mach-Zehnder modulators controls that only one of the feedback paths at most is active at any given time.

Figure 9

Temporal dynamics from short and long loop acquired for a pump current of $I=17.5\mathrm{mA}$. The time traces have been deskewed to overlap in time.

Figure 10

Correlation peaks in the double-delay-loop experiment. (a) Central peak structure of AC from original and replica coincide. (b) First delay echo $\alpha$ of original AC and CC coincide. (c) Central CC peak structure is symmetric and reveals consistency correlation ${\gamma }^2$ at its center. (d) First delay echo $\beta$ of replica AC coincides with CC where original and replica is exchanged.

Figure 11

Five selected peak values of correlation functions in dependence on the pump current $I$ of the laser: Consistency correlation ${\gamma }^2$ from CC, transformation correlation $\alpha$ from AC of original and CC (coinciding), and spurious correlation $\beta$ from AC of replica and CC (coinciding).

Figure 12

Examples of AC delay echoes (thick violet) within the corresponding analytic consistency ellipse Eq. (15) (light green). The first delay echo around the main delay time $\tau$ is calculated for the original signal yielding the $\alpha$ values, and from the replica signal yielding the corresponding $\beta$ values. Left panel: high consistency example. Right panel: low consistency example. Note the different scales. The peak structures tend to collapse approximately on a single line between identical ($\alpha =\beta$, black) and neutral line ($\beta =\alpha {\gamma }^2$, dark green). Delay echoes from other conditions (not shown) have a similar appearance. Despite experimental conditions, the boundaries of the ellipse are never violated.