## 3 Linear Stability Analysis
### Stable Analysis of the Model
 \tag{19}\]  
Substituting \(y_{n}(t)=ce^{i\alpha_{k}n+zt}\) and \(y_{n}{}^{{}^{\prime}}(t)=zc\epsilon^{i\alpha_{k}n+zt}\) into Formula (19) gives the simplified formula below:  
\[\big{(}e^{zt_{d}}-1\big{)}[ze^{zt_{d}}-f_{n}^{\rm v}+\rho f_{n}^{\rm\Delta v} \sum\limits_{q=1}^{Q}\alpha_{q}\Big{(}e^{-i\alpha_{k}q}-e^{-i\alpha_{k}(q-1) }\Big{)}\ -\ \mu_{n}^{\rm\alpha}\sum\limits_{q=1}^{Q}\beta_{q}ze^{-i\alpha_{k}q}]=t_{d}f_{ n}^{\rm s}\sum\limits_{q=1}^{Q}\varphi_{q}\Big{(}e^{-i\alpha_{k}}-e^{-i\alpha_{k}(q-1) }\Big{)} \tag{20}\]  
Step 3. Expand and organize the formula  
Further substituting \(e^{zt_{d}}=1+t_{d}z+\frac{t_{d}^{2}z^{2}}{2}+\ldots\) and \(e^{-i\alpha_{k}}=1-i\alpha_{k}+\frac{(i\alpha_{k})^{2}}{2}\) into Formula (20) gives:  
\[ \begin{array}{l}\bigg{(}t_{d}Z+\frac{t_{d}^{2}+Z^{2}}{2}\bigg{)}\bigg{[}Z \bigg{(}1+t_{d}Z+\frac{t_{d}^{2}Z^{2}}{2}\bigg{)}-f_{n}^{\rm\alpha}\\ +\rho f_{n}^{\rm\Delta v}\sum\limits_{q=1}^{Q}\alpha_{q}\bigg{(}\bigg{[}1-i \alpha_{k}+\frac{(i\alpha_{k})^{2}}{2}\bigg{]}^{q}-\bigg{[}1-i\alpha_{k}+\frac {(i\alpha_{k})^{2}}{2}\bigg{]}^{(q-1)}\bigg{)}\\ -\mu f_{n}^{\rm\alpha}\sum\limits_{q=1}^{Q}\beta_{q}Z\bigg{[}1-i\alpha_{k}+ \frac{(i\alpha_{k})^{2}}{2}\bigg{]}^{q}\bigg{]}\\ =t_{d}f_{n}^{\rm s}\sum\limits_{q=1}^{Q}\varphi_{q}\bigg{(}\bigg{[}1-i\alpha_{k}+ \frac{(i\alpha_{k})^{2}}{2}\bigg{]}^{q}-\bigg{[}1-i\alpha_{k}+\frac{(i\alpha_{k })^{2}}{2}\bigg{]}^{(q-1)}\bigg{)}\propto\end{array}  \tag{21}\]Expand z = \(z_{1}(i\alpha_{k})+z_{2}(i\alpha_{k})^{2}+\ldots\), insert it into Formula (21), and ignore high-order infinitesimals:  
\[ \begin{split}&\Bigg{(}t_{d}\Big{(}Z_{1}(i\alpha_{k})+Z_{2}(i \alpha_{k})^{2}\Big{)}+\frac{t_{d}^{2}\big{(}Z_{1}(i\alpha_{k})+Z_{2}(i\alpha_{ k})^{2}\big{)}^{2}}{2}\Bigg{)}\\ &*\Big{[}\Big{(}Z_{1}(i\alpha_{k})+Z_{2}(i\alpha_{k})^{2}\Big{)} \Bigg{(}1+t_{d}\Big{(}Z_{1}(i\alpha_{k})+Z_{2}(i\alpha_{k})^{2}\Big{)}+\frac{t_ {d}^{2}\big{(}Z_{1}(i\alpha_{k})+Z_{2}(i\alpha_{k})^{2}\big{)}^{2}}{2}\Bigg{)} -f_{n}^{\alpha}\\ &+\rho f_{n}^{\Delta\
u}\sum\limits_{q=1}^{Q}\alpha_{q}\Bigg{(} \bigg{[}1-i\alpha_{k}+\frac{(i\alpha_{k})^{2}}{2}\bigg{]}^{q}-\bigg{[}1-i \alpha_{k}+\frac{(i\alpha_{k})^{2}}{2}\bigg{]}^{(q-1)}\Bigg{)}\\ &-\mu f_{n}^{\alpha}\sum\limits_{q=1}^{Q}\beta_{q}\Big{(}Z_{1}(i \alpha_{k})+Z_{2}(i\alpha_{k})^{2}\Big{)}\bigg{[}1-i\alpha_{k}+\frac{(i\alpha _{k})^{2}}{2}\bigg{]}^{q}\big{]}\\ &=t_{d}f_{n}^{s}\sum\limits_{q=1}^{Q}\varphi_{q}\Bigg{(}\bigg{[}1 -i\alpha_{k}+\frac{(i\alpha_{k})^{2}}{2}\bigg{]}^{q}-\bigg{[}1-i\alpha_{k}+ \frac{(i\alpha_{k})^{2}}{2}\bigg{]}^{(q-1)}\Bigg{)}\end{split}  \tag{22}\]  
The first and second-order expressions of parameter Z can be derived by sorting as:  
\[Z_{1}=\sum\limits_{q=1}^{Q}\varphi_{q}\frac{f_{n}^{s}}{f_{n}^{s}} \tag{23}\]  
\[Z_{2}=\frac{\Big{[}\Big{(}1-\frac{t_{d}}{2}f_{n}^{\alpha}\Big{)}Z_{1}^{2}-f_{ n}^{\Delta\
u}\sum\limits_{i=1}^{Q}\alpha_{Fi}Z_{1}-Z_{1}^{2}\mu f_{n}^{s}\sum \limits_{i=1}^{q}\beta_{Fi}-f_{n}^{s}\sum\limits_{i=1}^{p}\varphi_{Fi}\Big{(}p -\frac{1}{2}\Big{)}\Big{]}}{f_{n}^{\alpha}}>0 \tag{24}\]  
Step 4. Derive the condition of traffic flow stability  
If \(Z_{2}>0\), the disturbed traffic flow will gradually tend to stability; else, a traffic disorder will result from the disturbance to the traffic flow.