## 3 Time-Delay Chaotic Synchronization Structure
If the condition \(s>0,w(s)>0\) is added to (1), then the zero solution of Equation (16) is consistently asymptotically stable. Previously, the time-lag phenomenon was often treated with a suppression approach in the design of chaotic synchronous systems, leading the system to an unstable periodic orbit. However, chaotic time-delay synchronous research has gradually developed to achieve complete reconfiguration of the chaotic states of two chaotic time-delay systems, which reduces the complexity and increases the reliability of the synchronized system [11]. Therefore, a multi-dimensional self-time-lagged chaotic synchronization method is found based on [PERSON]'s general function theory, and the error system is verified through numerical simulations. Finally, the relevant factors affecting the convergence speed of the self-time-lagged chaotic synchronization are studied. The four-dimensional Lorenz system is taken as the driving source for the self-time-lag synchronous system, and the time-lag response system is as follows  
\[\left\{ \begin{array}{l}\dot{\widetilde{x}}(t)=a(\widetilde{y}(t)-\widetilde{ x}(t))+p\widetilde{x}(t-\tau)+u_{1}(t)\\ \dot{\widetilde{y}}(t)=b\widetilde{x}(t)+c\widetilde{y}(t)+\widetilde{w}(t)- \widetilde{x}(t)\widetilde{z}(t)+u_{2}(t)\\ \dot{\widetilde{z}}(t)=\widetilde{x}(t)\widetilde{y}(t)-d\widetilde{z}(t)+u_ {3}(t)\\ \dot{\widetilde{w}}(t)=-h\widetilde{x}(t)+u_{4}(t)\end{array} \right. \tag{18}\]  
It is crucial to design a suitable synchronous controller for the chaotic synchronization of the two systems. Based on the theory of active control, the choice of controllers \(u_{1}(t),u_{2}(t),u_{3}(t),u_{4}(t)\) can achieve the global stability of the synchronized system. \[\left\{ \begin{array}{l}u_{1}(t)=-l_{1}e_{x}(t)-px(t-\tau)\\ u_{2}(t)=-l_{2}e_{y}(t)-x(t)z(t)+\widetilde{x}(t)\widetilde{z}(t)\\ u_{3}(t)=x(t)z(t)-\widetilde{x}(t)\widetilde{z}(t)-l_{3}e_{z}(t)\\ u_{4}(t)=-l_{4}e_{w}(t)\end{array} \right. \tag{19}\]  
where \(e_{x}=\widetilde{x}-x,e_{y}=\widetilde{y}-y,e_{z}=\widetilde{z}-z,e_{w}= \widetilde{w}-w\), the response system (18) is differenced from the four-dimensional Lorenz system to obtain the corresponding error system  
\[\left\{ \begin{array}{l}\dot{e}_{x}(t)=a(e_{y}(t)-e_{x}(t))+p\widetilde{x}(t -\tau)+u_{1}(t)\\ \dot{e}_{y}(t)=be_{x}(t)+ce_{y}(t)+e_{w}(t)-\widetilde{x}(t)\widetilde{z}(t)+ x(t)z(t)+u_{2}(t)\\ \dot{e}_{z}(t)=\widetilde{x}(t)\widetilde{z}(t)-x(t)z(t)-de_{z}(t)+u_{3}(t)\\ \dot{e}_{w}(t)=-he_{x}(t)+u_{4}(t)\end{array} \right. \tag{20}\]  
\(e_{i}(i=x,y,z,w)\) is called a chaotic synchronous error. According to the Lyapunov stability theorem, the synchronous error system (20) is asymptotically stable at the origin, which means the drive system (8) and the response system (18) are completely synchronized when the synchronous error tends to zero. The Lyapunov function is constructed\[V(t)=\frac{1}{2}(e_{x}^{2}(t)+e_{y}^{2}(t)+e_{z}^{2}(t)+e_{w}^{2}(t))+\int\limits_{- \tau}^{0}(e_{x}^{2}(t+\theta)+e_{y}^{2}(t+\theta)+e_{z}^{2}(t+\theta)+e_{w}^{2}(t +\theta))\text{d}\theta \tag{21}\]  
for all \(t\geq 0\), it is clearly shown that \(V(t)\) is a positive-definite function. Evaluating the time derivative of \(V(t)\) along the trajectory shown in the error system (4) gives  
\[ \begin{array}{ll}\dot{V}(t)&=e_{x}(t)\dot{e}_{x}(t)+e_{y}(t)\dot{e}_{y}(t)+e_{ z}(t)\dot{e}_{z}(t)+e_{z}(t)\dot{e}_{z}(t)+[(e_{x}^{2}(t)+e_{y}^{2}(t)+\\ &e_{z}^{2}(t)+e_{w}^{2}(t))-(e_{x}^{2}(t-\tau)+e_{y}^{2}(t-\tau)+e_{z}^{2}(t- \tau)+e_{w}^{2}(t-\tau))]\\ &=-(l_{1}+a-1)e_{x}^{2}(t)-(l_{2}-c-1)e_{y}^{2}(t)-(l_{3}+d-1)e_{z}^{2}(t)-(l_ {4}-1)e_{w}^{2}(t)\\ &+(a+b)e_{x}(t)e_{y}(t)+e_{y}(t)e_{w}(t)-he_{x}(t)e_{w}(t)+pe_{x}e(t-\tau)\\ &-(e_{x}^{2}(t-\tau)+e_{y}^{2}(t-\tau)+e_{z}^{2}(t-\tau)+e_{w}^{2}(t-\tau))] \end{array}  \tag{22}\]  
according to arithmetic-geometric mean inequality  
\[