## 2 Numerical Simulation
(**a**) Original \(x_{i}\) (**b**) after adding perturbation \(x_{1}\); (**c**) difference plot \(x_{1}\)-\(x\). The equilibrium point \(S\) of the system (1) is stable only if it meets \(\Delta_{1}=a+b-c>0\), \(\Delta_{2}=a^{2}b+ab^{2}-c-2 abc>0\), \(\Delta_{3}=2 abc>0\), and Equation (4) has three negative real roots. The equilibrium point \(S\) of the system (1) is unstable when \(\Delta_{1}=a+b-c<0\) or \(\Delta_{2}=a^{2}b+ab^{2}-c-2 abc<0\) or \(\Delta_{3}=2 abc<0\), and Equation (4) has a negative real part root and a pair of imaginary conjugate roots of the positive real part. Both equilibrium points are saddle focus points in three-dimensional space. System (1) is a dissipative system with a dispersion of  
\[ \begin{array}{c}\
abla V=\frac{\partial\dot{x}}{\partial x}+\frac{\partial \dot{y}}{\partial y}+\frac{\partial\dot{z}}{\partial z}+\frac{\partial\ddot{w} }{\partial w}=-(a-c+b)=-19\\ V(t)=V(0)e^{-19}\end{array}  \tag{7}\]  
Since \((a-c+b)>0\), the system (1) is always dissipative and converges in the exponential form of \(\dot{V}=e^{-(a-c+b)}\). Further, an initial volume \(V(0)\) converges to a volume element \(V(0)e^{-19}\) at time \(t\). Therefore, when \(t\rightarrow\infty\), each volume element containing the system's trajectory shrinks to zero at an exponential rate \(-(a-c+b)\). Therefore, all system trajectories will eventually be limited to a set of points with zero volume, and its asymptotic behavior will be fixed on an attractor. On the other hand, the linear feedback method extends the systemic dimension. The mathematical model of a four-dimensional structured chaotic system based on the Lorenz system is as follows  
\[\left\{ \begin{array}{l}\dot{x}(t)=a(y(t)-x(t))\\ \dot{y}(t)=bx(t)+cy(t)-x(t)z(t)+w(t)\\ \dot{z}(t)=x(t)y(t)-dz(t)\\ \dot{w}(t)=-hx(t)\end{array} \right. \tag{8}\]  
there are only two nonlinear terms in system (8), \(a\), \(b\), \(c\), \(d\), and \(h\) are the control parameters, while the values \(a=35\), \(b=7\), \(c=12\), \(d=3\) and \(h=5\), and the system enters a chaotic state, the trajectories of chaotic attractors are shown in Figure 3.  
In reality, there is often a time lag between the state variables of a dynamical system. The evolutionary trend of a system is related to the current state of the system and the state at a certain time in the past, thus evolving a time-delay dynamical system. We introduce delay time in the mathematical model of the four-dimensional Lorenz disturbance term \(px(t-\tau)\) to form a new chaotic time-delay system. \[\left\{ \begin{array}{l}\dot{x}(t)=a(y(t)-x(t))+px(t-\tau)\\ \dot{y}(t)=bx(t)+cy(t)-x(t)z(t)+w(t)\\ \dot{z}(t)=x(t)y(t)-dz(t)\\ \dot{w}(t)=-hx(t)\end{array} \right. \tag{9}\]  
The variables \(p\) and \(\tau\) in Equation (9) represent the hysteresis control parameter and the hysteresis time of the time-delay system, respectively, which can be appropriately assigned to drive the newly constructed system into a chaotic state. The power spectrum of the trajectory \(x(t)\) and sequence of chaotic attractors for this system are shown in Figure 4, which are obtained by picking \(p=3\), \(\tau=1.4\) ms and initial values \([-20;0;0;0]\). Figure 3: Phase diagram of a four-dimensional chaotic attractor. (**a**) \(z\)-\(y\)-\(x\) diagram; (**b**) \(x\)-\(z\)-\(w\) diagram; (**c**) \(x\)-\(y\) plane phase diagram; (**d**) \(x\)-\(z\) plane phase diagram. The sensitivity of the initial values is one of the most important properties of chaos, which makes long-term prediction of the system impossible. Figure 5 shows the time domain waveform of the initial function \(x(t)\) of the chaotic time-delay model combined with the difference between the \(10^{-8}\) disturbance and the original series as a function of time \(t\). Figure 4: Phase diagram of a time-delay four-dimensional Lorenz chaotic attractor. (**a**) \(x\)-\(z\) plane phase diagram; (**b**) \(x\)-\(y\)-\(z\) plane phase diagram; (**c**) the power spectrum of the variable \(x(t)\). The results show that the differences curve into a random vibration after some time.