## 4 Numerical Examples  
In this section, we will apply a typical example from other literature to verify the correctness and validity of our results. All numerical calculations are performed by MATLAB. Moreover, the analysis shows that our technique is a reliable approach to solve the nonlinear dynamics of gene expression systems. **Example 1**. : _The typical example comes from [2], and it should be pointed out that this example is a special case in the study of this article_  
\[ \begin{array}{rcl}\dot{m}(t)&=&\frac{k}{1+(p(t-\tau)/40)^{4}}-0.16m\\ \dot{p}(t)&=&m-0.1p\end{array}  \tag{28}\]\(\mu=0.16\) and \(d=0.1\) denote the degradation rates of mRNA and protein, respectively, where \(p_{0}=40\) is a reference concentration of protein \(n=4\)._  
_The associating characteristic polynomial (6) is_  
\[\lambda^{2}+0.26\lambda+0.016+Ke^{-\tau\lambda}\]  
_In the following, we will discuss the special case when \(k=1\). Lemma 1 tells us that_  
\[\omega_{0}=0.0990\]  
_and Theorem (1) leads us to the critical delay_  
\[\tau_{0}=18.2674\]  
_and_  
\[Re\lambda^{{}^{\prime}}(\tau_{0})=0.00092\]  
_Our simulation diagram gives the time delay stability interval, through the previous theoretical calculation, and the interval we calculated is consistent with the interval in the simulation, and both aspects can verify each other._  
_We take the root trajectory Figure 1, and we find that, at \(\tau<18.2674\), the system is stable since the eigenroots all have negative real parts. In addition, a pair of pure imaginary roots crosses the imaginary axis at \(\tau_{0}=18.2674\), resulting in the unstable oscillation of the system, that is, Hopf bifurcation. Thus, there exists only one delay interval \([0,18.2674]\), for the system of gene expression._  
_The corresponding parameters are calculated as follows: Lemmas 2 and 3 give the adjoint vectors as_  
\[q(\theta)=\left[ \begin{array}{c}0.1000+0.0990j\\ 1\end{array} \right]e^{i\omega_{0}\theta}\]  
\[q^{*}(\theta)=\left[ \begin{array}{c}1\\ 0.2418+0.1192j\end{array} \right]e^{i\omega_{0}\theta}\]  
_Associating with Lemma 4, we obtain_  
\[ \begin{array}{l}H_{2}=0.000111,H_{3}=0.000024\\ g_{20}=-0.000011+0.00029j\\ g_{11}=0.000142-0.000254j\\ g_{02}=-0.000242+0.00016j\end{array} \]  
Figure 1: The rightmost root loci versus delay \(\tau\). Subsequently, Lemma 5 describes that \(y_{20}\) and \(y_{11}\) are such that_  
\[ \begin{array}{rcl}y_{20}^{(2)}(-\tau)&=&0.001752+0.004121j\\ y_{11}^{(2)}(-\tau)&=&-0.001292\end{array} \]  
_Thus, we have_  
\[g_{21}=-0.000182-0.000049j\]  
_Therefore, it gives the bifurcation parameters_  
\[\mu_{2}=0.099129,\ \tau_{2}=0.003557\text{ and }\beta_{2}=-0.000182\]  
_Theorem 2 tells us that the Hopf bifurcation occurs at \(\tau=18.2674\), and the direction of bifurcation is supercritical._  
_In addition. we draw stable and unstable phase trajectoru in Figures 2 and 3._  
Furthermore, we took a representative point inside the stability interval, \(\tau=14\); this time delay represents that the system is stable. We drew a phase diagram according to this time lag, and the track of the phase diagram finally converged to the origin, which also shows that the system is stable. Then, it shows that we obtained this time lag, and it is indeed stable. Figure 3: The two phase trajectories with delay in unstable intervals. Figure 2: The stable phase trajectory with delay in a stable interval. _At the same time, we took another point from the unstable interval. For \(\tau=21\) in Figure 4, we chose two different initial states. It is easy to see that both of the two phase trajectories are convergent to the bifurcating periodic solution. It means that both phase trajectories are divergent, and both are further and further away from the equilibrium point. In addition, this means that the equilibrium points are unstable._  
_Finally, we took the bifurcation point, a special point, and found that the graph converges to the limit ring.