## 3 Liner Stability Analysis  
Many macroscopic and microscopic traffic models apply the stability analysis method for the theoretical investigation of the effects of different external conditions on the traffic flow. Through the linear stability analysis, it is possible to calculate the stability interval of the modified car-following model. It can also assess the influence of traffic density and the acceleration of the preceding car on traffic flow kinetics when parameters are changed. \(N\) vehicles are assumed to be distributed on a single-lane ring road, with circular road length \(L\) and steady initial fleet state, then all vehicles move with similar car spacing \(h\) and fixed optimal velocity \(V(h)\). Thus, steady-state solution can be obtained as:  
\[x_{n}^{0}(t)=ln+V(h)t,\ h=L/N, \tag{7}\]  
where \(x_{n}^{0}(t)\) is the location of car \(n\) at time \(t\). By linearizing the original nonlinear system, the stability condition of this steady-state solution is obtained. First, \(y_{n}(t)\) is assumed as a minor deviation from the steady-state \(x_{n}^{0}(t)\), the position of the vehicle after the disturbance is added as:  
\[x_{n}(t)=x_{n}^{0}(t)+y_{n}(t), \tag{8}\]  
By the substitution of Equation (7) and Equation (8) into Equation (5), followed by linearization gives:  
\[ \begin{array}{ll}\frac{d^{2}y_{n}(t)}{dt^{2}}=&\alpha\{(1-p)V[h+\Delta y_{n} (t)]+pV[\frac{1}{m}\sum\limits_{j=0}^{m-1}(h+\Delta y_{n+j}(t))]-(V(h)+\frac{ dy_{n}(t)}{dt})\}\\ &\hskip 113.811024 pt+\beta\frac{d^{2}y_{n+1}(t)}{dt^{2}}+\lambda\frac{d\Delta y _{n}(t)}{dt}\end{array} , \tag{9}\]  
Introducing finite increment formula into Equation (9), the equation is rewritten as:  
\[\frac{d^{2}y_{n}(t)}{dt^{2}}=\alpha\{(1-p)V^{\prime}(h)\Delta y_{n}(t)+pV^{ \prime}(h)[\frac{1}{m}\sum\limits_{j=0}^{m-1}\Delta y_{n+j}(t)]-\frac{dy_{n}( t)}{dt}\}+\beta\frac{d^{2}y_{n+1}(t)}{dt^{2}}+\lambda\frac{d\Delta y_{n}(t)}{dt}, \tag{10}\]  
where \(V^{\prime}(h)=\frac{dV(\Delta x_{n})}{d\Delta x_{n}}\) and \(\Delta x_{n}=h\), which is the optimal velocity function's first order partial derivative at the headway \(h\). The solution of Equation (10) is obtained by expanding the Fourier series with \(y_{n}(t)=A\ exp\ (ikn+zt)\) as an orthonormal set. The equation transformed is:  
\[z^{2}=\alpha\{(1-p)V^{\prime}(h)(e^{ik}-1)+pV^{\prime}(h)[\frac{1}{m}\sum \limits_{j=0}^{m-1}(e^{ik(j+1)}-e^{ikj})]-z\}+\beta z^{2}e^{ik}+\lambda z(e^{ ik}-1) \tag{11}\]  
Expanding \(z\) into a power series form \(z=z_{1}(ik)+z_{2}(ik)^{2}+\dots\) and using Taylor series to expand the exponent \(e^{x}=1+x+x^{2}/2+\dots\), then substituting it into Equation (11), we have a new power series equation that contains only first and second-order terms after rounding off the high-order terms, as follows:  
\[\alpha(V^{\prime}(h)-z_{1})(ik)+(\alpha(\frac{1}{2}V^{\prime}(h)(1+(m-1)p)-z_ {2})+\lambda z_{1}+(\beta-1)z_{1}^{2})(ik)^{2}=0, \tag{12}\]When the first and second coefficients of the power series are 0, Equation (12) is established. Therefore, the solution of \(z_{1}\), \(z_{2}\) is derived as:  
\[\left\{ \begin{array}{c}z_{1}=V^{\prime}(h)\\ z_{2}=\frac{V^{\prime}(h)[a[1+(m-1)p]+2\lambda+2V^{\prime}(h)(\beta-1)]}{2 \alpha}\end{array} \right., \tag{13}\]  
Since \(y_{n}\)(t) is the interference from steady traffic flow, when the amplitude of \(y_{n}\)(t) is increased with time evolution, the car-following state is unstable. Therefore, for small disturbances with long wavelengths, the oscillation of this model will be reduced if \(z_{2}>0\). Thus, the uniform traffic flow tends to the steady state in the condition that:  
\[V^{\prime}(h)<\frac{\alpha[1+(m-1)p]+2\lambda}{2(1-\beta)}, \tag{14}\]  
In other words, the critical stability curve is given by:  
\[\alpha=\frac{2[(1-\beta)V^{\prime}(h)-\lambda]}{1+(m-1)p}, \tag{15}\]  
When \(p=0\), \(\beta=0\) or \(m=1\), \(\beta=0\), stable condition is similar to that obtained from the FVD model, as Equation (16). Therefore, in this paper, the FVD model is assumed to be a special case of the DAVD model.