Zero Lag Synchronization of Chaotic Systems with Time Delayed Couplings

Zero-lag synchronization (ZLS) between chaotic units, which do not have self-feedback or a relay unit connecting them, is experimentally demonstrated for two mutually coupled chaotic semiconductor lasers. The mechanism is based on two mutual coupling delay times with certain allowed integer ratios, whereas for a single mutual delay time ZLS cannot be achieved. This mechanism is also found numerically for mutually coupled chaotic maps where its stability is analyzed using the Schur-Cohn theorem for the roots of polynomials. The symmetry of the polynomials allows only specific integer ratios for ZLS. In addition, we present a general argument for ZLS when several mutual coupling delay times are present.

Figure 1

Experimental setup: two semiconductor lasers coupled by two bidirectional couplings with different optical path lengths. LD, laser diode; PD, photodiode; BS, beam splitter; ND, neutral density filter; CC, corner cube.

Figure 2

Recording of the time dependent intensities of two zero-lag synchronized lasers (solid lines) and the calculated cross correlation of the two lasers (black circles). Notice the desynchronization periods associated with power dropouts (LFFs).

Figure 3

Cross correlation coefficient for two chaotic units which are mutually coupled via two time-delayed couplings with ${\tau }_1$ and ${\tau }_2$. Dots: Lang-Kobayashi simulation with ${\tau }_1=10\mathrm{ns}$ and ${\tau }_2$ is varied, $\sigma =50\mathrm{n}{\mathrm{s}}^{-1}$. Triangles: Experimental data with same parameters as simulation. Squares: Data from simulation with Bernoulli units with $a=0.45$, $b=c=0.3$; see Eq. (2). Error bars mark the standard deviation from several simulations of the Lang-Kobayashi equations.

Figure 4

A 10 ns recording of the time dependent intensity of two semiconductor lasers (thin blue and thick red). (a) ${\tau }_2/{\tau }_1=2$, (b) ${\tau }_2/{\tau }_1=\frac{3}{2}$, (c) ${\tau }_2/{\tau }_1=\frac{4}{3}$, (d) ${\tau }_2/{\tau }_1=\frac{5}{3}$. To demonstrate the slow decorrelation of the intensities with increasing time we show in panel (a) (dashed line) the intensity of the laser after a delay of $4{\tau }_1$. Corresponding correlation values can be seen in Fig. 3.

Figure 5

ZLS for two units coupled by two time-delayed couplings (${\tau }_1,{\tau }_2$). Only the chaotic region is shown. Region with horizontal lines, ${\tau }_2/{\tau }_1=2$; vertical lines, ${\tau }_2/{\tau }_1=3/2$; diagonal lines, ${\tau }_2/{\tau }_1=5/2$.