## 2 Numerical Simulation
Its maximum difference can reach 31.7147, indicating that the trajectory of chaotic vibration is extremely sensitive to the initial value. The nonperiodic nature of the time domain waveform makes the system hard to predict. That is a consequence of positive Lyapunov exponents acting on the time-lag system (9). Figure 6 describes the Lyapunov exponential graph as it varies with the steps when the initial values are [\(-20\); \(0\); \(0\); \(0\)], control parameters \(a=35\), \(b=7\), \(c=12\), \(d=3\), \(h=5\), \(p=3\), and delay-time \(\tau=1.4\) ms. There are two positive Lyapunov exponents when the evolution of the Lyapunov exponent is stable, respectively \(\lambda_{1}=0.6607\), \(\lambda_{2}=0.0524\). The maximum Lyapunov exponent of the chaotic attractor is measured to be 0.6607. Thus, the time-delay chaotic attractor constructed by introducing a perturbation term is more complex than the classical three-dimensional Lorenz attractor in dynamic behavior. This property is particularly valuable in secure communication. Figure 5: \(x_{1}\)-\(x\) difference trajectory diagram. Figure 6: Lyapunov exponents of the time-delay system. Analyze the stability of its system equilibrium point  
\[\left\{ \begin{array}{l}a(y(t)-x(t))+px(t-\tau)=0\\ bx(t)+cy(t)-x(t)z(t)+w(t)=0\\ x(t)y(t)-dz(t)=0\\ -lx(t)=0\\ \end{array} \right. \tag{10}\]  
At the equilibrium point, the solution of Equation (10) is a constant independent of time t. It is a constant state solution \(x=x_{0}\), \(y=y_{0}\), \(z=z_{0}\), \(w=w_{0}\). It cannot give a track line that changes with time, and there can be only a one-track line through each point on the phase plane except the equilibrium point, implying that the tracks cannot intersect. The solution gives the system only one equilibrium point \(S_{0}=(0,0,0,0)\). Stability analysis of this equilibrium point is performed by giving the system a small perturbation \(\delta x,\delta y,\delta\hat{z},\delta w\) to move out of equilibrium. Assuming that the solution of the system (9) is \(x=x_{0}+\delta x\), \(y=y_{0}+\delta y\), \(z=z_{0}+\delta z\), \(w=w_{0}+\delta w\), and substituting them into the original system (9) and expanding the right-hand side of the equation to the linear term in Taylor's series form, we obtain at the equilibrium point  
\[\left\{ \begin{array}{l}\delta\dot{x}=f_{1}(x_{0},y_{0},z_{0},w_{0})+\frac{ \partial f_{1}}{\partial x}\delta x+\frac{\partial f_{1}}{\partial y}\delta y +\frac{\partial f_{1}}{\partial z}\delta z+\frac{\partial f_{1}}{\partial w} \delta w\\ \delta\dot{y}=f_{2}(x_{0},y_{0},z_{0},w_{0})+\frac{\partial f_{2}}{\partial x} \delta x+\frac{\partial f_{2}}{\partial y}\delta y+\frac{\partial f_{2}}{ \partial z}\delta z+\frac{\partial f_{2}}{\partial w}\\ \delta\dot{z}=f_{3}(x_{0},y_{0},z_{0},w_{0})+\frac{\partial f_{3}}{\partial x} \delta x+\frac{\partial f_{3}}{\partial y}\delta y+\frac{\partial f_{3}}{ \partial z}\delta z+\frac{\partial f_{3}}{\partial w}\delta w\\ \delta\dot{w}=f_{4}(x_{0},y_{0},z_{0},w_{0})+\frac{\partial f_{4}}{\partial x} \delta x+\frac{\partial f_{4}}{\partial y}\delta y+\frac{\partial f_{4}}{ \partial z}\delta z+\frac{\partial f_{4}}{\partial w}\end{array} \right. \tag{11}\]  
Equation (11) is a linear equation with a perturbation of \(\delta x\delta y\delta z\delta w\). Jacobin matrix can be obtained at the right end, recorded as \(J\)  
\[J=\left( \begin{array}{cccc}\frac{\partial f_{1}}{\partial x}&\frac{\partial f _{1}}{\partial y}&\frac{\partial f_{1}}{\partial z}&\frac{\partial f_{1}}{ \partial w}\\ \frac{\partial f_{2}}{\partial x}&\frac{\partial f_{2}}{\partial y}&\frac{ \partial f_{2}}{\partial z}&\frac{\partial f_{2}}{\partial w}\\ \frac{\partial f_{3}}{\partial x}&\frac{\partial f_{3}}{\partial y}&\frac{ \partial f_{3}}{\partial z}&\frac{\partial f_{3}}{\partial w}\\ \frac{\partial f_{4}}{\partial x}&\frac{\partial f_{4}}{\partial y}&\frac{ \partial f_{4}}{\partial z}&\frac{\partial f_{4}}{\partial w}\end{array}  \right)=\left( \begin{array}{cccc}-a&a&0&0\\ b-z&c&-x&1\\ y&x&-d&0\\ -h&0&0&0\end{array} \right)=\left(