## 2 Numerical Simulation  
The famous American meteorologist, Professor [PERSON], first described Lorenz systems in the 1963 [45]. This paper adopts a three-dimensional Lorenz chaotic system as the following set of third-order differential equations. \[\left\{ \begin{array}{l}\dot{x}(t)=a(y(t)-x(t))\\ \dot{y}(t)=-x(t)z(t)+cy(t)\\ \dot{z}(t)=x(t)y(t)-bz(t)\end{array} \right. \tag{1}\]  
where \(x(t)\), \(y(t)\) and \(z(t)\) are the state variables of the system, \(a\), \(b\) and \(c\) are the control parameters of the system. When the system parameters \(a=36\), \(b=3\), and \(c=20\), the system is in a chaotic state, and the trajectory of the system's solution in phase space is obtained. Figure 1a shows the three-dimensional phase diagram on the \(x\)-\(y\)-\(z\), and Figure 1b-d shows the phase diagram on two planes. Figure 1b-d show the trajectory of chaotic attractors on a two-dimensional plane. Chaotic motion is locally unstable, while the whole is stable. The initial value of a chaotic system is affected by minor disturbances. The distance between the points on the orbit and the corresponding points in the original orbit increases exponentially over a period of time. Under the influence of the boundedness of the chaotic motion, the distance between the two points oscillates randomly again. Further, the initial sensitivity experiment was carried out with the initial values chosen as \((-3,\ -4,\ 14)\) and the step size chosen as \(h=0.01\). The calculated curve of \(x\) with time \(t\) is shown in Figure 2a, Figure 2b shows the waveform \(x_{1}(t)\) after adding a perturbation \(10^{-8}\) to the initial function \(x(t)\), and Figure 2c shows the difference between them. Figure 1: Phase diagram of a three-dimensional chaotic attractor. (**a**) \(x\)-\(y\) plane phase diagram; (**b**) \(x\)-\(z\) plane phase diagram; (**c**) \(x\)-\(z\) plane phase diagram; (**d**) three-dimensional diagram. Let \(\dot{x}=\dot{y}=\dot{z}=0\) gives  
\[\left\{ \begin{array}{l}a(y(t)-x(t))=0\\ -x(t)z(t)+cy(t)=0\\ x(t)y(t)-bz(t)=0\end{array} \right. \tag{2}\]  
calculate the equilibrium points of system (1) and get the result that if \(bc\leq 0\), system (1) has only one equilibrium point \(S_{0}(0,0,0)\), and if \(bc>0\), system (1) has three equilibrium points \(S_{-}(-\sqrt{bc},-\sqrt{bc},c)\), \(S_{0}(0,0,0)\), \(S_{+}(\sqrt{bc},\sqrt{bc},c)\). Linearizing the system (1) on the equilibrium point \(S_{0}\) to obtain three eigenvalues: \(\lambda_{1}=-a,\lambda_{2}=c,\lambda_{3}=-b\). If \(c>0\), the origin is a saddle point in three-dimensional space; if \(c<0\), the origin is the only equilibrium point. The following is an analysis of the non-zero balance points of the system (1) so that the two non-zero balance points \(S_{-},S_{+}\) are \(S(x_{s},y_{s},z_{s})\). Linearize the system (1) on two non-zero equilibrium points, and calculate the Jacobian matrix  
\[\left| \begin{matrix}-a-\lambda&a&0\\ -z_{s}&c-\lambda&-x_{s}\\ y_{s}&x_{s}&-b-\lambda\end{matrix} \right|=0 \tag{3}\]  
The characteristic polynomial is obtained  
\[\lambda^{3}+(a+b-c)\lambda^{2}+(ab-ac-bc+x_{s}^{2}+az_{s})\lambda+(ax_{s}^{2}+ abz_{s}+ax_{s}y_{s}-abc)=0 \tag{4}\]  
substitute the values \(S_{-}(-\sqrt{bc},-\sqrt{bc},c)\), \(S_{+}(\sqrt{bc},\sqrt{bc},c)\) of the non-zero equilibrium point  
\[\lambda^{3}+(a+b-c)\lambda^{2}+ab\lambda+2 abc=0 \tag{5}\]  
Assume the coefficients in the characteristic polynomial are \(a_{0}=1,a_{1}=a+b-c,a_{2}=ab,a_{3}=2 abc\). Since the sign of the real part of the eigenvalues of the derived linear system can be determined by using the Routh Hurwitz criterion. It can be inferred whether the corresponding nonlinear system is stable. According to the Routh-Hurwitz criterion, there are  
\[ \begin{array}{l}\Delta_{0}=a_{0}=1>0\\ \Delta_{1}=a_{1}=a+b-c\\ \Delta_{2}=\left|\begin{matrix}a_{1}&a_{0}\\ a_{3}&a_{2}\end{matrix}\right|=\left|\begin{matrix}a+b-c&1\\ 2 abc&ab\end{matrix}\right|=a^{2}b+ab^{2}-c-2 abc\\ \Delta_{3}=a_{3}=2 abc\end{array}  \tag{6}\]  
Figure 2: Time course diagram.