Complete chaotic synchronization in mutually coupled time-delay systems

Complete chaotic synchronization of end lasers has been observed in a line of mutually coupled, time-delayed system of three lasers, with no direct communication between the end lasers. The present paper uses ideas from generalized synchronization to explain the complete synchronization in the presence of long coupling delays, applied to a model of mutually coupled semiconductor lasers in a line. These ideas significantly simplify the analysis by casting the stability in terms of the local dynamics of each laser. The variational equations near the synchronization manifold are analyzed, and used to derive the synchronization condition that is a function of parameters. The results explain and predict the dependence of synchronization on various parameters, such as time delays, strength of coupling and dissipation. The ideas can be applied to understand complete synchronization in other chaotic systems with coupling delays and no direct communication between synchronized subsystems.

Figure 1

A schematic showing how three lasers are coupled in a line. The outer two lasers (circles) are identical, while the middle laser (square) is detuned from the rest.

Figure 2

(Color online) Top: Intensity of laser 1 vs laser 2. Bottom: Laser 1 vs laser 3. Straight line indicates complete synchronization of outer lasers. $\tau =30$, $ϵ=\sqrt{0.001}$, ${\delta }_1={\delta }_2=6.5ϵ$.

Figure 3

(Color online) (a) Sum of Lyapunov exponents as a function of dissipation, $ϵ$, for $\tau =120$. (b) Corresponding correlations between outer lasers, $\tau =120$. (c) Sum of Lyapunov exponents vs $ϵ$, for $\tau =240$. (d) Corresponding correlations between outer lasers, $\tau =240$. In all cases, $a_1=a_2=2$, $b_1=b_2=1$, ${\delta }_1={\delta }_2=0.2$, $\beta =0.5$.

Figure 4

(Color online) (a) Numerically computed sum of Lyapunov exponents as a function of delay, $\tau$. (b) Corresponding correlations of outer lasers. (c) Correlations of the middle and outer lasers, shifted by the delay time to maximize correlations. $ϵ=\sqrt{0.001}$, ${\delta }_1={\delta }_2=7.5ϵ$, $\beta =0.5$.

Figure 5

(Color online) (a) Sum of Lyapunov exponents as a function of coupling strength, ${\delta }_1={\delta }_2$, for $\tau =60$. (b) $\tau =120$. $ϵ=\sqrt{0.001}$, $\beta =0.5$.

Figure 6

(Color online) Correlations corresponding to Fig. 5. (a) Correlation between the middle and one of the outer lasers, $\tau =60$. (b) Correlations of outer lasers, $\tau =60$. (c) Correlation between the middle and one of the outer lasers, $\tau =120$. (d) Correlations of outer lasers, $\tau =120$. Outer lasers synchronize for greater range of coupling strength as the delay is increased. The middle and the outer lasers show little correlation for all values of the coupling strengths.