## 5 Simulation of a Self-Synchronous Circuit for a Four-Dimensional Time-Delay Lorenz System
Then, based on the Lyapunov stability theory, the analysis of the Lyapunov functional of the chaotic dynamical error system (35) yields  
\[V(e(t))=\frac{1}{2}e^{T}(t)e(t)+\int\limits_{-\tau}^{0}e^{T}(t+\theta)e(t+ \theta)d\theta \tag{36}\]then the derivative of \(V\) for time \(t\) is  
\[ \begin{split}\dot{V}(e(t))&=e^{T}(t)\dot{e}(t)+e^{T}(t )e(t)-e^{T}(t-\tau)e(t-\tau)\\ &=e^{T}(t)L_{1}e(t)-e^{T}(t-\tau)L_{2}e(t-\tau)\end{split}  \tag{37}\]  
where \(L_{1},L_{2}\) are positive definite matrices, both represent the coefficient matrices after Lyapunov functional analysis. By selecting the feedback gain matrix \(Q\) appropriately, it is possible to make the system matrix \(L_{1}\) negative and thus keep the derivative of \(V(e(t))\) negative. In this case, the error system (33) is globally asymptotically stable near the origin, meaning that for any initial condition, \(\lim\limits_{t\to\infty}\lVert e(t)\rVert=0\). We introduce the coupled synchronous scheme into a four-dimensional time-delay Lorenz chaotic system to achieve coupled chaotic synchronization of two time-lagged hyperchaotic systems with the same initial values, prove the corresponding asymptotic conditions for synchronous convergence, and innovatively propose a coupled synchronous control circuit that can be verified by simulation to demonstrate that the method is effective. The proposed four-dimensional Lorenz system (8) is used as the driving system for the coupled synchronization and is written in matrix form as follows  
\[\dot{X}(t)=AX(t)+f(X(t),t) \tag{38}\]  
where \(X(t)=\left[X_{1}(t)\,\ X_{2}(t)\,\ \cdots\,\ X_{4}(t)\right]^{T}\),  
\[A= \begin{bmatrix}-a&a&0&0\\ b&c&0&1\\ 0&0&-d&0\\ -h&0&0&0\end{bmatrix} ,\ f(X(t),t)= \begin{bmatrix}0\\ -x(t)z(t)\\ x(t)y(t)\\ 0\end{bmatrix} \]  
the constructed time-delay chaotic system (9) is taken as a coupled synchronous response system as follows  
\[\dot{\widetilde{X}}(t)=A\widetilde{X}(t)+f(\widetilde{X}(t),t)+P\widetilde{X} (t-\tau)+Q(X(t)-\widetilde{X}(t)) \tag{39}\]  
let us denote the error system  
\[e(t)=X(t)-\widetilde{X}(t)= \begin{bmatrix}e_{1}(t)\\ e_{2}(t)\\ e_{3}(t)\\ e_{4}(t)\end{bmatrix} ,\ e_{i}(t)=X_{i}(t)-\widetilde{X}_{i}(t)\]  
the error system of the coupled system (39) and system (38) is  
\[\dot{e}(t)=Ae(t)-P\widetilde{X}(t-\tau)-f(\widetilde{X}(t),t)+f(X(t),t)-Qe(t) \tag{40}\]  
where  
\[ \begin{split}& f(X(t),t)-f(\widetilde{X}(t),t)=\begin{bmatrix} 0\\ -X(t)Z(t)+\widetilde{X}(t)\widetilde{Z}(t)\\ X(t)Y(t)-\widetilde{X}(t)\widetilde{Y}(t)\\ 0\end{bmatrix}\\ &=\begin{bmatrix}0&0&0&0\\ -Z(t)&0&-\widetilde{X}(t)&0\\ Y(t)&\widetilde{X}(t)&0&0\\ 0&0&0&0\end{bmatrix}\begin{bmatrix}e_{1}(t)\\ e_{2}(t)\\ e_{3}(t)\\ e_{4}(t)\end{bmatrix}=Ke(t)\end{split}  \tag{41}\]  
The synchronization of the coupled system (39) with the system (38) is achieved by simply satisfying certain conditions on the parameters of the coupling function so that its error system is asymptotically stable as time tends to infinity, allowing the nonlinear coupling and synchronization of two uniform chaotic systems with different initial conditions but the same structure. In the coupled system (39), the matrix of functions of the state variables is taken  
\[Q= \begin{bmatrix}q_{1}&-Z(t)&Y(t)&0\\ 0&q_{2}&0&0\\ 0&0&q_{3}&0\\ 0&0&0&q_{4}\end{bmatrix}  \tag{42}\]  
where \(q_{i}\geq 0\) (\(i=1\,,\,2\,,\,3\,,\,4\)) are the parameters to be determined. We obtain the error system (40) as  
\[\dot{e}(t)=Ae(t)+(K-Q)e(t)-P\widetilde{X}(t-\tau) \tag{43}\]  
by calculating, we get  
\[K-Q= \begin{bmatrix}-q_{1}&Z(t)&-Y(t)&0\\ -Z(t)&-q_{2}&-\widetilde{X}(t)&0\\ Y(t)&\widetilde{X}(t)&-q_{3}&0\\ 0&0&0&-q_{4}\end{bmatrix}  \tag{44}\]  
then we can write  
\[\frac{1}{2}\Big{[}(K-Q)+(K-Q)^{T}\Big{]}=diag(-q_{1}\,,\,-q_{2}\,,\,-q_{3}\,, \,-q_{4}) \tag{45}\]  
We construct the Lyapunov function \(V(e)=\frac{1}{2}e^{T}(t)e(t)+\int_{-\tau}^{0}\big{(}e^{T}(t+\theta)e(t+\theta )\big{)}\mathrm{d}\theta\) and find its derivative for time \(t\) as  
\[