## 3 Time-Delay Chaotic Synchronization Structure  
In recent years, chaos has been used extensively in secure communication. Chaos synchronization has long been studied in the context of low-dimensional chaotic systems with only one positive Lyapunov exponent, and synchronized communication schemes using such low-dimensional chaos are easily deciphered by prediction and reconstruction methods. In contrast, high-dimensional hyperchaotic communication systems based on multiple positive Lyapunov exponents have better secrecy than low-dimensional chaotic systems. However, the structure is more complex, making chaos synchronization more difficult. Time-delay chaotic systems are infinite-dimensional systems with more complex dynamical behavior, capable of generating positive Lyapunov exponents in more dimensions. The study of their synchronous systems is of great value in practical applications, making them a key focus in the study of synchronization of hyperchaotic systems [14]. The Lyapunov first and second methods are tools for studying the stability of general continuous dynamical systems described by ordinary differential equations. The first method determines the stability of a system based on the eigenvalues of differential equations. In contrast, the second method uses Lyapunov functions to analyze the global stability of the system. For time-delay chaos, the Lyapunov exponent of the system is related to a function on the initial period. Time-delay chaos is an infinite-dimensional system. It is more difficult to analyze the accuracy of the Lyapunov exponent if it is solved using local linearization. Using the Lyapunov first method in analyzing time-delay chaos is not advisable. Therefore, the extension of the Lyapunov second method to continuous dynamical systems with time lags in simultaneous method determination is called Lyapunov generalization. Set \(C[-r,0]\) to be the space of all continuous functions from \([-r,0]\) to \(R^{n}\), and \(r>0\), for any \(\phi\in C[-r,0]\), whose parameterization is \(\|\phi\|=\sup\limits_{-r\leq\theta\leq 0}\|\phi|\), where \(\|\cdot\|\) is the parameterization in \(R^{n}\). When \(x(t)\) is a continuous function on \([-r,T](0\leq T\leq+\infty)\), define \(x_{t}(\theta)=x(t+\theta)\), for any \(t\in[0,T]\), \(\theta\) takes all values on \([-r,0]\). A general system of continuous differential dynamics with time delays is as follows  
\[\dot{x}=F(t,x_{t}) \tag{16}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Eienvalue** & \(\mathbf{\Delta}_{1}\) & \(\mathbf{\Delta}_{2}\) & \(\mathbf{\Delta}_{3}\) \\ \hline \(\lambda^{4}\) & \(1\) & \(-596\) & \(525\) \\ \(\lambda^{3}\) & \(26\) & \(-350\) & \(0\) \\ \(\lambda^{2}\) & \(\frac{26\times(-596)-1\times(-350)}{26}\approx-582.5\) & \(\frac{26\times 525-1\times 0}{26}=525\) & \(0\) \\ \(\lambda^{1}\) & \(\frac{-582.5\times(-350)-26\times 525}{-5825}\approx 373.4\) & \(0\) & \(0\) \\ \(\lambda^{0}\) & \(525\) & \(0\) & \(0\) \\ \hline \end{tabular}
\end{table} 
Table 1: System stability analysis. where \(F:R\times C\to R^{n}\) is continuous, and \(F(t,0)=0\), ensuring the existence and uniqueness of solution \(x(t,t_{0},\phi)\) for all initial value \((t_{0},\phi)\), Equation (16), denoted \(x(t)=x(t,t_{0},\phi)\). Let \(V:R\times C\to R\) be a continuous function and define  
\[\dot{V}=\dot{V}(t,\varphi)=\varlimsup_{h\to 0^{+}}\frac{1}{h}(V(t+h,x_{t+h})-V(t, \phi)) \tag{17}\]  
The Lyapunov functional method for stability of time-delay systems is given. Set \(u(s),v(s),w(s):R^{+}\to R^{+}\) to be continuous and non-decreasing functions. When \(s>0\), \(u(s)>0\), \(v(s)>0\), \(u(0)=v(0)=w(0)=0\), then  
1. A zero solution \(x=0\) of Equation (16) is consistently stable if the function \(V:R\times C\to R\) exists, such as, \(u(\|\phi(0)\|)\leq V(t,\phi)\leq v(\|\phi\|)\), \(\dot{V}(t,\phi)\leq-w(\|\phi(0)\|)\). 2. The zero solution of Equation (16) is uniformly bounded if the condition \(\varliminf_{s\to+\infty}u(s)=+\infty\) is added to (1). 3.