## 5 Simulation of a Self-Synchronous Circuit for a Four-Dimensional Time-Delay Lorenz System
\begin{array}{c}\dot{V}(e)=e^{T}(t)\dot{e}(t)+[e^{2}(t)-e^{2}(t-\tau)]\\ =e^{T}(t)Ae(t)+e^{T}(t)(K-Q)e(t)+[e^{2}(t)-e^{2}(t-\tau)]\\ =e^{T}(t)\frac{A+A^{T}}{2}e(t)+e^{T}(t)Qe(t)+[e^{2}(t)-e^{2}(t-\tau)]\\ =e^{T}(t)\mathrm{Re}(t)+[e^{2}(t)-e^{2}(t-\tau)]\end{array}  \tag{46}\]  
the calculation yields  
\[R= \begin{bmatrix}-a-q_{1}&\frac{a+b}{2}&0&-\frac{b}{2}\\ \frac{a+b}{2}&c-q_{2}&0&0\\ 0&0&-d-q_{3}&0\\ -\frac{b}{2}&0&0&-q_{4}\end{bmatrix}  \tag{47}\]  
To make \(R\) negative, we must meet the following conditions for  
\[ \begin{array}{c}(1)\ \Delta_{1}=-a-q_{1}<0\\ (2)\ \Delta_{2}=(a+q_{1})(q_{2}-c)-\frac{(a+b)^{2}}{4}>0\\ (3)\ \Delta_{3}=-(d+q_{3})\Delta_{2}<0\\ (4)\ \Delta_{4}=-q_{4}\Delta_{3}+\frac{b^{2}}{4}(c-q_{2})(-d-q_{3})>0\end{array}  \tag{48}\]  
We compute the principal subdivisions of the matrix \(R\) at each level of order and know that  
\[q_{1}>-35,q_{2}>453,q_{3}>-3,q_{4}>0, \tag{49}\]  
At this point, \(R\) is negative, thus the derivative of \(V(e(t))\) is also negative, so the error system (42) is globally asymptotically stable at the origin, \(\lim\limits_{t\to\infty}\lVert e(t)\rVert=0\) for any initial condition. Based on the chaotic time-delay system in the simulation experiment in Figure 10, we used Multisim software to design the univariate coupled time-lag chaotic oscillator synchronous circuit shown in Figure 12. The solution involves controlling the synchronization of two constructive time-delay chaotic systems with similar evolutionary laws by appropriately driving the system variable \(Y\) back to the response system, thus controlling the system to synchronize its variable coupled system. Figure 12: Constructed chaotic time-delay synchronous circuit. We choose the chaotic system (8) as the driving system so that the response (controlled) system is  
\[\left\{ \begin{array}{l}\dot{x}_{1}(t)=a(y_{1}(t)-x_{1}(t))+px_{1}(t-\tau)\\ \dot{y}_{1}(t)=bx_{1}(t)+cy_{1}(t)-x_{1}(t)z_{1}(t)+w_{1}(t)+q(y(t)-y_{1}(t))\\ \dot{z}_{1}(t)=x_{1}(t)y_{1}(t)-dz_{1}(t)\\ \dot{w}_{1}(t)=-hx_{1}(t)\end{array} \right. \tag{50}\]  
By choosing the appropriate control gain \(q\), we can adjust the response system (50) without affecting the drive system (8) so that the state of the system (50) converges to the system (8) and is eventually fully synchronized. We show the synchronous control circuit in Figure 13 with the variable \(Y\) of the drive system (8) and the variable \(Y1\) of the response system (50) at the two inputs, and the controlled feedback value \(Y-Y1\) at the output. To illustrate the synchronous effect of this control circuit, we can use the nodal voltage method to derive the state equation \(\widetilde{Y}-\widetilde{Y}1\) in front of the resistor \(R_{34}\) as follows  
\[V_{\widetilde{Y}-\widetilde{Y}1}=-\frac{R_{31}}{R_{29}}Y+\frac{R_{31}R_{32}}{ R_{30}R_{33}}Y_{1} \tag{51}\]  
Different control gain \(q\) can be produced by tuning the resistance values of \(R_{29}\), \(R_{30}\), \(R_{31}\), \(R_{32}\), and \(R_{34}\), as shown in Table 3.  
We compare the synchronous effect of this synchronous control circuit for different control gains and observe the system variable \(Y-Y1\) on an oscilloscope. The synchronous phase diagrams are presented in Figure 14. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(R_{29}\), \(R_{30}\), \(R_{31}\), \(R_{32}\) (k\(\Omega\)) & & [10, 10, 10] & \\ \hline \(R_{34}\) (k\(\Omega\)) & 100 & 300 & 500 & 700 \\ \hline control gain \(q\) & 10 & 3.3 & 2 & 1.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Synchronous control gain. Figure 13: Synchronous control circuit. Through numerical simulations, we have found that the two time-lagged chaotic systems can only be synchronized gradually when the control gain \(q>2.5\), and that the chaos synchronous effect is proportional to the value of the gain. The value of the gain \(q\) can be adjusted independently by changing the value of the resistor \(R_{34}\). \[q=\frac{1}{R_{34}\cdot C} \tag{52}\]  
The threshold value corresponding to the control gain is taken as \(R_{34}=400\) k\(\Omega\).