## 5 Stability in Mean and Robustness with Regard to Disturbances
\right]=0 \tag{27}\]  
which is obtained by means of the perturbations \(v=v_{0}+\Delta v\) and \(\overline{\alpha}\overline{\gamma}=\overline{\alpha}\overline{\gamma}\) for \((\alpha,\gamma)\in\{z,u,y,x\}\). The insertion of the perturbations into the averaged Equations (24) and (25) and linearization in the \(\Delta\)-terms yields the fifth order stability Equation (27) with the characteristic equation \(\Delta(\lambda)=\)\(A_{0}\lambda^{5}+A_{1}\lambda^{4}+\cdots A_{5}=0\). The determinant of Equation (27) yields  
\[A_{5}=\frac{(z_{0}\Omega)^{2}Dv_{0}^{4}[v_{0}^{4}-6(1-2D^{2})v_{0}^{2}+5]}{ \left(1-v_{0}^{2}\right)^{2}+(2 Dv_{0})^{2}}. \tag{28}\]  
According to the [PERSON] criterion, \(A_{5}<0\) determines divergence that gives the boundary of monotonous instability. Obviously, the stability condition \(A_{5}>0\) coincideswith the positive slope condition of the speed driving force characteristic calculable by differentiating Equation (12) with respect to the travel speed \(v\) that gives  
\[\frac{d}{dv}\left(\frac{f\Omega}{c}\right)=(z_{0}\Omega)^{2}v^{4}\frac{v^{4}-6 \left(1-2D^{2}\right)v^{2}+5}{\left[\left(1-v_{0}^{2}\right)^{2}+\left(2 Dv \right)^{2}\right]^{2}}>0\;.\]  
This result coincides with Equation (28) except that the positive definite denominator in Equation (28) is squared. Hence, the negative slope condition and the instability in mean lead to the same instability boundary. This is plausible and physically explainable, as already done in the yellow area in Figure 2a. The instable speed range vanishes with increasing damping. In addition to the stability behavior, the robustness of the limit cycle calculation with regard to disturbances is of interest. When disturbances are generated by brief shocks, the eigenvalues of the stability matrix in Equation (27) must be calculated. They determine the growth behavior with which the disturbances increase or decrease, respectively. In the case of stationary disturbances, noise models are introduced. Figure 4a shows a stochastic limit cycle obtained for the quarter car model in the case that the angle motion on the sinusoidal road form is perturbed by additive noise given by  
\[d\phi_{\tau}=-V_{\tau}d\tau+\sigma dW_{\tau},\qquad\qquad\Delta W_{n}=\sqrt{ \Delta\tau}\;N_{n}. \tag{29}\]  
In Equation (29), capital letters with index \(\tau\) denote set functions [21; 22] dependent on time. Noise is generated by normally distributed numbers \(N_{n}\) with zero mean [23]. The stochastic angle perturbation takes into account that the road surface is no longer sinusoidal but more realistically irregular and noisy with bounded realizations. For small noise intensities \(\sigma\), this leads to response realizations which are bounded, as well. Initial results are shown in Figure 4a where trajectories of a stochastic limit cycle are plotted in the phase plane of travel velocity and acceleration scaled by 0.3 and shifted by the applied driving force. The realizations are calculated by means of Equations (24)-(26) where the polar angle in Equation (26) is replaced by Equation (29). The applied damping is given by \(D=0.2\), the driving force by \(f\Omega/c=0.6\), the road level by \(z_{0}\Omega=0.9\), and the noise intensity by \(\sigma=0.03\). The Euler scheme is applied with the time step \(\Delta\tau=10^{-4}\). The mean value of the shifted acceleration and true velocity is marked by a yellow triangle on the red curve of the speed driving force characteristic, indicating that the mean acceleration is vanishing and the mean travel speed coincides with Equation (12). The comparison with the double-periodic yellow limit cycle in Figure 2b shows that its sharp line is widened to a bundle of nonperiodic realizations, the boundaries of which are double periodic with  
Figure 4: **(a)** Limit cycle flow of a bundle of irregular trajectories with double-periodic side limits when the acceleration is plotted against the true vehicle speed.