## 5 Simulation of a Self-Synchronous Circuit for a Four-Dimensional Time-Delay Lorenz System  
Researchers have gradually realized the value of chaos synchronization and its application in chaos theory research in recent years, such as adaptive, self-activating control, and backstepping methods. The coupled synchronous control method was used in the research of [PERSON] and [PERSON] in 1990. Then [PERSON] conducted an in-depth analysis and study of the coupled synchronization of [PERSON]'s circuits, theoretically proving that as long as the coupling between two chaotic systems is strong enough, chaotic synchronization can be achieved. However, most of the coupled synchronous schemes that have been proposed are limited to chaotic systems with only one positive Lyapunov exponent. In contrast, relatively few studies have been carried out to realize coupled synchronization of multi-dimensional chaotic systems or even hyperchaotic systems. Extending the coupled synchronous scheme to hyperchaotic synchronous systems would be a research trend [14]. There are many basic methods for determining the existence of chaotic synchronization, such as the Routh-Hurwitz stability discrimination method, the method for solving the optimal solution of linear inequalities based on Lyapunov stability, and the Lyapunov  
Figure 11: Two-dimensional planar projection of chaotic time-delay system. (a) \(V_{y}\)-\(V_{z}\) (0.5 V/div, 1 V/div); (b) \(V_{x}\)-\(V_{y}\) (1 V/div, 2 V/div). function method [56]. We will choose the Lyapunov function method to determine the existence of a coupled synchronous system model. The dynamical equations of the chaotic system take the form  
\[\dot{X}(t)=AX(t)+f(X(t),t) \tag{29}\]  
where \(X\in R^{n}\) is the n-dimensional state vector of the system, \(A\) is the n-dimensional constant matrix, and \(f\) is a smooth nonlinear continuous function, and adding the time-delay linear perturbation term gives  
\[\dot{X}(t)=AX(t)+f(X(t),t)+PX(t-\tau) \tag{30}\]  
\(P\) is an n-dimensional constant matrix and \(\tau\) is the time lag constant of the system. By choosing an appropriate time delay \(\tau\) and linear perturbation \(P\), the system (30) can be left in a chaotic state. We take the system (29) as the driving system and obtain the response system as  
\[\dot{\widetilde{X}}(t)=A\widetilde{X}(t)+f(\widetilde{X}(t),t)+P\widetilde{X }(t-\tau)+U(t) \tag{31}\]  
where \(U(t)=(U_{1}(t)\,\ U_{2}(t)\ \cdots U_{n}(t))\) is the synchronous controller. For an initial value, the drive system (29) is said to have achieved synchronization with the response system (31) if it is such that  
\[\lim_{t\rightarrow\infty}\lVert X(t)-\widetilde{X}(t)\rVert=0 \tag{32}\]  
the following error system is obtained by subtracting the general chaotic system (29) from the constructed time-lag chaotic system (31)  
\[\dot{e}(t)=\dot{X}(t)-\dot{\widetilde{X}}(t)=Ae(t)-P\widetilde{X}(t-\tau)-f( \widetilde{X}(t),t)+f(X(t),t)-Qe(t) \tag{33}\]  
The response system (31) can be adjusted without affecting the drive system (29) to achieve a coupled synchronization of the drive system (31) and the response system (33) with a suitably selected control gain \(Q\). From the median theorem, it follows that  
\[f(X(t),t)-f(\widetilde{X}(t),t)=Ke(t) \tag{34}\]  
where \(K\) is the Jacobi matrix of \(f\). In Equation (33), the control matrix \(Q=diag(q_{1}\,\ q_{2}\ \cdots\ q_{n})\) is referred to as the feedback gain matrix  
\[\dot{e}(t)=(A+K-Q)e(t)-P\widetilde{X}(t-\tau)=A_{f}(t)e(t)-P\widetilde{X}(t-\tau) \tag{35}\]  
At this point, Equation (35) is a time-varying system of chi-square linear equations. An appropriate feedback gain matrix \(Q\) can be chosen so that the eigenvalues of the matrix \(A_{f}(t)\) all have negative real parts, allowing equation (32) to be implemented. The drive system (29) and response system (31) will be coupled and synchronized.