Zero-lag chaos synchronization properties in a hierarchical tree-type network consisting of mutually coupled semiconductor lasers

Abstract

Chaos synchronization properties in a novel hierarchical tree-type optical network consisting of semiconductor lasers (SLs) are investigated numerically. In such network, each child node is asymmetrically mutually coupled with the parent node. Zero-lag synchronization is found among the nodes belonging to the same layer, while the nodes belonging to different layers are not synchronized, which is denoted as hierarchical chaos synchronization (HCS). The effects of injection strength, asymmetrical injection parameters, bias currents and frequency detuning on the performance of HCS and chaotic complexity are studied in detail. Our results show that high-quality HCS can be obtained under moderate parameters. The stability of the network is also investigated systematically. Further, we explore the robustness of HCS by considering an asymmetrical network. Finally, output bit streams are generated from each SL at a tunable rate of up to about 10 Gbps with verified randomness. These results are enlightening to the multi-user chaotic communication and the synchronization mechanism in biological neuronal networks.

Appendices

Appendix A: the method of calculating the largest Lyapunov exponent

In calculating the largest Lyapunov exponent, the linearized form of the coupled Lang–Kobayashi equations is required. For simplicity, the electric-field amplitude and phase of the coupled Lang–Kobayashi equations are derived from the complex electric-field equations, and k is used instead of $k_{{\mathrm{II,III}}}$: (take ${{\mathrm{SL}}_4}$ as an example)

$$\begin{aligned} \frac{{\mathrm{d}\mathbf{E }_{4}(t)}}{{\mathrm{d}t}}= & {} \frac{{1}}{2}\left[ \frac{g(N_4(t)-N_0)}{1+s{|\mathbf{E }_4(t)|}^2} - {\tau _p}\right] \mathbf{E }_4(t)\nonumber \\&+\,k\mathbf{E }_2(t - {\tau _d})\exp (-i\omega \tau _d) \end{aligned}$$

(A.1)

The complex electric fields $\mathbf{E }_{2}(t)$ and $\mathbf{E }_{4}(t)$ can be presented by the real values of the amplitudes $E_{2}(t)$ and $E_{4}(t)$ and the real values of the phases $\varPhi _{2}(t)$ and $\varPhi _{4}(t)$:

$$\begin{aligned} \mathbf{E }_{2}(t)= & {} E_{2}(t)\exp (i\varPhi _2(t)) \end{aligned}$$

(A.2)

$$\begin{aligned} \mathbf{E }_{4}(t)= & {} E_{4}(t)\exp (i\varPhi _4(t)) \end{aligned}$$

(A.3)

By substituting Eqs. (A.2) and (A.3) into Eq. (A.1) and using Euler’s formula $\exp (-ix)=\cos (x)-i\sin (x)$, the real and imaginary parts can be separated as:

$$\begin{aligned} \frac{{\mathrm{d}E_{4}(t)}}{{\mathrm{d}t}}= & {} \frac{{1}}{2}\left[ \frac{g(N_4(t)-N_0)}{1+sE_4^2(t)} - {\tau _p}\right] E_4(t) \nonumber \\&+\, kE_2(t - {\tau _d})\cos (\varTheta _{42}(t)) \end{aligned}$$

(A.4)

$$\begin{aligned} \frac{{\mathrm{d}\varPhi _{4}(t)}}{{\mathrm{d}t}}= & {} \frac{{\alpha }}{2}\left[ \frac{g(N_4(t)-N_0)}{1+sE_4^2(t)} - {\tau _p}\right] \nonumber \\&-\, k\frac{E_2(t - {\tau _d})}{E_4(t)}\sin (\varTheta _{42}(t)) \end{aligned}$$

(A.5)

$$\begin{aligned} \frac{{\mathrm{d}N_{4}(t)}}{{\mathrm{d}t}}= & {} J_4-\frac{{N_{4}(t)}}{\tau _s}-\frac{{g(N_4(t)-N_0)}}{1+sE_4^2(t)}E_4^2(t) \end{aligned}$$

(A.6)

$$\begin{aligned} \varTheta _{42}= & {} \omega \tau _d+\varPhi _4(t)-\varPhi _2(t-\tau _d) \end{aligned}$$

(A.7)

Then, the variables that describe the small linear deviations from the oscillatory trajectory on the attractor can be expressed by the linearized form:

$$\begin{aligned} \varDelta x={{\mathrm{Ja}}}\cdot \mathrm{d}x \end{aligned}$$

(A.8)

where Ja denotes the Jacobian matrix of the considered system. Note that for dynamical systems with time delay, the time delay has an extra degree of freedom, because the dynamics are governed by all the initial conditions of the variables within the delay time $\tau$ [39, 45]. It is thus very important to consider all the contributions from the time-delayed variables at different delay times. According to Ref. [45], a continuous equation of the small separations $\delta x$ which represent a difference between two functions can be derived as:

$$\begin{aligned} \frac{{\mathrm{d}\delta x}}{{\mathrm{d}t}}=\frac{{\partial F(x,x_\tau )}}{\partial x}\delta x+\frac{{\partial F(x,x_\tau )}}{\partial x_\tau }\delta x_\tau \end{aligned}$$

(A.9)

In the network we have seven nodes of three dimensions each, namely $\mathrm{d}E$, $\mathrm{d}\varPhi$, $\mathrm{d}N$, and two time-delayed dimensions $\mathrm{d}E_\tau$, $\mathrm{d}\varPhi _\tau$. So the Jacobian matrix has a size of $3 \times 7$ (21) rows and $3 \times 7+2 \times 7$ (35) columns. Then, the Jacobian matrix can be obtained by calculating the partial derivatives of each separation variable at time t. The largest Lyapunov exponent is calculated from the above linearized equations [44, 45]:

$$\begin{aligned} \lambda _{\max }=\frac{{1}}{L\tau } \sum \nolimits _{k=1}^{L}\log \frac{{\parallel \delta _ {x(k)}\parallel }}{\parallel \delta _{x(k-1)}\parallel } \end{aligned}$$

(A.10)

where $\tau$ is the delay time, and L is the steps repeated in the algorithm. In the continuous time system, the norm $\parallel \cdot \parallel$ that describes the deviation from t to $t+\tau$ is defined as the Euclidean distance of the linearized variables:

$$\begin{aligned} D(t)=\sqrt{\sum \limits _{k=0}^{M-1}\left( \delta _x^2(t+kh)+\delta _{x\tau }^2(t+kh) \right) } \end{aligned}$$

(A.11)

where $M=\tau /h$. The norm that describes the deviation from $t+\tau$ to $t+2\tau$ can be expressed as

$$\begin{aligned} D(t+\tau )=\sqrt{\sum \limits _{k=0}^{M-1}\left( \delta _x^2(t+\tau +kh)+\delta _{x\tau }^2(t+\tau +kh) \right) } \end{aligned}$$

(A.12)

The ratio of two norms that describes the expansion rate of the separation is

$$\begin{aligned} d_j=\frac{{D(t+\tau )}}{D(t)} \end{aligned}$$

(A.13)

The normalization of the linearized variables is necessary as to keep the linearity of the stability analysis, namely to make sure a small deviation from the trajectory [39]:

$$\begin{aligned} \delta _{x,x\tau }(t+\tau +kh)=\frac{{\delta _{x,x\tau }(t+\tau +kh)}}{D(t+\tau )} \end{aligned}$$

(A.14)

Hence, according to Eqs. (A.10–A.14), we can calculate the largest Lyapunov exponent, as presented in Fig. 9.

Appendix B: the method of calculating the conditional Lyapunov exponent

In calculating the conditional Lyapunov exponent $\lambda _c$ that describes the synchronization stability of two SLs, the linearized form of the deviation variables is required. For simplicity, the stability of synchronization between ${{\mathrm{SL}}_4}$ and ${{\mathrm{SL}}_5}$ is analyzed as an example. The deviation variables are defined as:

$$\begin{aligned} \varDelta _E= & {} E_4(t)-E_5(t) \end{aligned}$$

(B.1)

$$\begin{aligned} \varDelta _{\varPhi }= & {} \varPhi _{4}(t)-\varPhi _{5}(t) \end{aligned}$$

(B.2)

$$\begin{aligned} \varDelta _N= & {} N_4(t)-N_5(t) \end{aligned}$$

(B.3)

For simplicity, we assume the gain saturation coefficient s equals to 0. Then, the differential equations of ${{\mathrm{SL}}_4}$ and ${{\mathrm{SL}}_5}$ (nodes 4 and 5) can be written as:

$$\begin{aligned} \frac{{\mathrm{d}E_{4}(t)}}{{\mathrm{d}t}}= & {} \frac{{1}}{2}\left[ g(N_4(t)-N_0) - {\tau _p}\right] E_4(t) \nonumber \\&+\, kE_2(t - {\tau _d})\cos (\varTheta _{42}(t)) \end{aligned}$$

(B.4)

$$\begin{aligned} \frac{{\mathrm{d}E_{5}(t)}}{{\mathrm{d}t}}= & {} \frac{{1}}{2}\left[ g(N_4(t)-N_0) - {\tau _p}\right] E_5(t) \nonumber \\&+\, kE_2(t - {\tau _d})\cos (\varTheta _{52}(t)) \end{aligned}$$

(B.5)

$$\begin{aligned} \frac{{\mathrm{d}\varPhi _{4}(t)}}{{\mathrm{d}t}}= & {} \frac{{\alpha }}{2}\left[ g(N_4(t)-N_0) - {\tau _p}\right] \nonumber \\&-\, k\frac{E_2(t - {\tau _d})}{E_4(t)}\sin (\varTheta _{42}(t)) \end{aligned}$$

(B.6)

$$\begin{aligned} \frac{{\mathrm{d}\varPhi _{5}(t)}}{{\mathrm{d}t}}= & {} \frac{{\alpha }}{2}\left[ g(N_5(t)-N_0) - {\tau _p}\right] \nonumber \\&-\, k\frac{E_2(t - {\tau _d})}{E_5(t)}\sin (\varTheta _{52}(t)) \end{aligned}$$

(B.7)

$$\begin{aligned} \frac{{\mathrm{d}N_{4}(t)}}{{\mathrm{d}t}}= & {} J-\frac{{N_{4}(t)}}{\tau _s}-g(N_4(t)-N_0)E_4^2(t) \end{aligned}$$

(B.8)

$$\begin{aligned} \frac{{\mathrm{d}N_{5}(t)}}{{\mathrm{d}t}}= & {} J-\frac{{N_{5}(t)}}{\tau _s}-g(N_5(t)-N_0)E_5^2(t) \end{aligned}$$

(B.9)

$$\begin{aligned} \varTheta _{ij}= & {} \omega \tau _d+\varPhi _i(t)-\varPhi _j(t-\tau _d) \end{aligned}$$

(B.10)

What we finally need are the linearized equations:

$$\begin{aligned} \begin{aligned} \frac{{\mathrm{d}\varDelta _{E}(t)}}{{\mathrm{d}t}}= & {} [\bullet ]\varDelta _E(t)+[\bullet ]\varDelta _{\varPhi }(t)+[\bullet ]\varDelta _N(t)\\ \frac{{\mathrm{d}\varDelta _{\varPhi }(t)}}{{\mathrm{d}t}}= & {} [\bullet ]\varDelta _E(t)+[\bullet ]\varDelta _{\varPhi }(t)+[\bullet ]\varDelta _N(t)\\ \frac{{\mathrm{d}\varDelta _{N}(t)}}{{\mathrm{d}t}}= & {} [\bullet ]\varDelta _E(t)+[\bullet ]\varDelta _{\varPhi }(t)+[\bullet ]\varDelta _N(t) \end{aligned} \end{aligned}$$

(B.11)

The linearized equations can be obtained by using Eqs. (B.1–B.10):

$$\begin{aligned} \left( \begin{array}{ccccccc} \dfrac{{\mathrm{d}\varDelta _E(t)}}{{\mathrm{d}t}}\\ \dfrac{{\mathrm{d}\varDelta _{\varPhi }(t)}}{{\mathrm{d}t}}\\ \dfrac{{\mathrm{d}\varDelta _N(t)}}{{\mathrm{d}t}}\\ \end{array} \right)= & {} \left( \begin{array}{ccccccc} \dfrac{{\partial f_{\varDelta _E}}}{{\partial \varDelta _E}} &{}\quad \dfrac{{\partial f_{\varDelta _E}}}{{\partial \varDelta _{\varPhi }}} &{}\quad \dfrac{{\partial f_{\varDelta _E}}}{{\partial \varDelta _N}} \\ \dfrac{{\partial f_{\varDelta _{\varPhi }}}}{{\partial \varDelta _E}} &{}\quad \dfrac{{\partial f_{\varDelta _{\varPhi }}}}{{\partial \varDelta _{\varPhi }}} &{}\quad \dfrac{{\partial f_{\varDelta _{\varPhi }}}}{{\partial \varDelta _N}} \\ \dfrac{{\partial f_{\varDelta _N}}}{{\partial \varDelta _E}} &{}\quad \dfrac{{\partial f_{\varDelta _N}}}{{\partial \varDelta _{\varPhi }}} &{}\quad \dfrac{{\partial f_{\varDelta _N}}}{{\partial \varDelta _N}} \\ \end{array} \right) \nonumber \\&\times \,\left( \begin{array}{ccccccc} \varDelta _E(t)\\ \varDelta _{\varPhi }(t)\\ \varDelta _N(t)\\ \end{array} \right) \end{aligned}$$

(B.12)

The first term at the right side of Eq. (B.12) is the Jacobian matrix of the linearized system:

$$\begin{aligned} \left( \begin{array}{ccccccc} \dfrac{{1}}{2}\!\left[ g(N_4(t)-N_0)-{\tau _p}\right] &{}\quad -\, kE_2(t - \tau _d)\sin (\varTheta _{52}(t)) &{}\quad \dfrac{{1}}{2}g E_5(t)\\ k\dfrac{E_2(t - {\tau _d})}{E_5^2(t)}\!\sin (\varTheta _{52}(t)) &{}\quad -\, k \dfrac{E_2(t -{\tau _d})}{E_5(t)}\!\cos (\varTheta _{52}(t)) &{}\quad \dfrac{{\alpha g}}{2}\\ 2gE_5(t)\left( N_5(t)-N_0 \right) &{}\quad 0 &{}\quad \dfrac{{1}}{-\tau _s}-gE_5^2\\ \end{array} \right) \end{aligned}$$

(B.13)

Then, the conditional Lyapunov exponent $\lambda _c$ can be calculated using similar methods as introduced in “Appendix A.” The result is shown in Fig. 10.