## 3 Time-Delay Chaotic Synchronization Structure
\begin{array}{ll}\dot{V}(t)&\leq-(l_{1}+a-1)e_{x}^{2}(t)-(l_{2}-c-1)e_{y}^{2 }(t)-(l_{3}+d-1)e_{z}^{2}(t)-(l_{4}-1)e_{w}^{2}(t)\\ &+(a+b)e_{x}(t)e_{y}(t)+e_{y}(t)e_{w}(t)-he_{x}(t)e_{w}(t)+\frac{p^{2}}{2}e_{x} ^{2}+e_{x}^{2}(t-\tau)\\ &-(e_{x}^{2}(t-\tau)+e_{y}^{2}(t-\tau)+e_{z}^{2}(t-\tau)+e_{w}^{2}(t-\tau))]\\ &\leq-(l_{1}+a-1-p^{2}/4)e_{x}^{2}(t)-(l_{2}-c-1)e_{y}^{2}(t)-(l_{3}+d-1)e_{z} ^{2}(t)\\ &-(l_{4}-1)e_{w}^{2}(t)+(a+b)e_{x}(t)e_{y}(t)+e_{y}(t)e_{w}(t)-he_{x}(t)e_{w}(t) \end{array}  \tag{23}\]  
Assume \(e^{T}(t)=(e_{x},e_{y},e_{z},e_{w})\), taking on the structure of the quadratic form  
\[\dot{V}(t)\leq-e^{T}( \begin{pmatrix}l_{1}+a-1-\frac{p^{2}}{4}&-a-b&0&h\\ 0&l_{2}-c-1&0&-1\\ 0&0&l_{3}+d-1&0\\ 0&0&0&l_{4}-1\end{pmatrix} e(t) \tag{24}\]  
By [PERSON]'s sufficient condition, the error system (4) is asymptotically stable when the derived function \(\dot{V}(t)\) is negative as the control parameters \(l_{1}>1-a+p^{2}/4\), \(l_{2}>c+1\), \(l_{3}>1-d\), \(l_{4}>1\), achieving the chaotic time-delay synchronization of the drive system and the response system. When delay time \(\tau=1.4\)ms, control parameters \(p=3\), \(l_{1}=-1,l_{2}=24,l_{3}=2,l_{4}=4\), the initial values of the drive Equation (8) are \([-17;7;18;15]\), the initial values of response Equation (18) are \([8;-9;6;20]\), and the synchronous error curves can be obtained in Figure 7.  
Through the curves \(e_{x},e_{y},e_{z},e_{w}\), the error system converges to zero in a relatively short time under active control, which shows that the time-lag synchronous system can achieve chaotic synchronization rapidly with good robustness and stability. The effects of the control parameter \(l\), the initial value of the drive response system, and the time lag \(\tau\) on the speed t of synchronous convergence are given. Table 2 indicates that the speed of convergence on the delay synchronous system is more sensitive to changes in the control parameters. In contrast, the initial value of the drive response system and time delay has less influence on the convergence rate, and convergence speed slows down significantly with the control parameter \(l\) increase. For further research, it is expected that the proposed time-delay system state can be adaptively adjusted by improving the control algorithm to ensure that the system can work dynamically in an optimal state. Figure 7: Diagram of the error synchronous process for time-delay systems. (**a**) Variable \(x\) curve; (**b**) variable \(y\) curve; (**c**) error curves. \begin{table}
\begin{tabular}{c c c c c c c c} \hline Initial & & & & & & & \\ values & & & & & & & \\ \([-19;\,6;\,20;\,16;\,10;\,-10;\,8;\,22]\) & & & & & & \\ \([-17;\,7;\,18;\,15;\,8;\,-9;\,6;\,20]\) \\ \hline \(l\) & \([-3;\,14;\,-1;\,2]\) & \([-3;\,14;\,-1;\,2]\) & \([-3;\,14;\,-1;\,2]\) & \([-1;\,24;\,2;\,4]\) & \([-1;\,24;\,2;\,4]\) \\ \hline \(\tau/(\text{ms})\) & \(1.4\) & \(2.4\) & \(1.4\) & \(2.4\) & \(1.4\) & \(2.4\) & \(1.4\) & \(2.4\) \\ \hline \(t/(\text{s})\) & \(12.5\) & \(12.7\) & \(11.9\) & \(10.5\) & \(14.7\) & \(14.8\) & \(13.5\) & \(14.0\) \\ \hline \end{tabular}
\end{table} 
Table 2: System synchronous convergence time comparison table.