## 4 Numerical Results
### Case 2: Study with a Random Friction Coefficient and Young's Modulus of the Disc  
In the second case study, the friction coefficient and Young's modulus of the disc are considered to be random. Utilizing the random variables, they are modeled as  
\[ \begin{array}{ll}\mu&=0.5+0.5\;\zeta_{1}\\ E_{\rm disc}&=125\times 10^{9}(1+0.2\zeta_{2})\end{array}  \tag{22}\]  
where \(\zeta_{1}\,,\zeta_{2}\) are uniform random variables within the interval \([-1,1]\). The purpose of the second example is to validate the combined nonstationary kriging and support vector machine method. The reference results for the real part of the first three unstable eigenvalues are presented in Figure 10. In obtaining the reference, a 2D linear sample grid (\(40\times 40\)) is considered. The obtained results show that the sensitivity of the bifurcation friction coefficient to the disk's random variable is large at the first eigenvalue, making the response surface more complex. \begin{table}
\begin{tabular}{l c c c c c c c} \hline  & **Re(\(\lambda_{1}\))** & **Re(\(\lambda_{2}\))** & **Re(\(\lambda_{3}\))** & **Im(\(\lambda_{1}\))** & **Im(\(\lambda_{2}\))** & **Im(\(\lambda_{3}\))** & **\(\Sigma\)**Re(\(\lambda_{4}\))** & **Im(\(\lambda_{4}\))** \\ \hline
0 th & \(3.35\times 10^{-2}\) & \(8.47\times 10^{-3}\) & \(1.21\times 10^{-2}\) & \(5.93\times 10^{-5}\) & \(3.24\times 10^{-5}\) & \(1.19\times 10^{-4}\) & \(5.42\times 10^{-2}\) & \(2.10\times 10^{-4}\) \\
1 st & \(3.35\times 10^{-2}\) & \(3.06\times 10^{-2}\) & \(2.57\times 10^{-2}\) & \(9.54\times 10^{-5}\) & \(8.85\times 10^{-5}\) & \(1.17\times 10^{-4}\) & \(8.98\times 10^{-2}\) & \(3.01\times 10^{-4}\) \\
2 nd & \(4.14\times 10^{-2}\) & \(2.00\times 10^{-2}\) & \(2.60\times 10^{-2}\) & \(5.74\times 10^{-5}\) & \(1.52\times 10^{-4}\) & \(1.36\times 10^{-4}\) & \(8.75\times 10^{-2}\) & \(3.45\times 10^{-4}\) \\
3 rd & \(4.10\times 10^{-2}\) & \(1.57\times 10^{-2}\) & \(2.34\times 10^{-2}\) & \(1.07\times 10^{-4}\) & \(3.81\times 10^{-5}\) & \(2.16\times 10^{-4}\) & \(8.02\times 10^{-2}\) & \(3.61\times 10^{-4}\) \\ \hline \end{tabular}
\end{table} 
Table 4: Error results for the three unstable eigenvalue pairs using the nonstationary kriging with nine samples. The trend functions with polynomial order from 0 th to 3 rd are applied. Figure 10: Reference results for the real part of three unstable eigenvalue pairs. A 2D linear sample grid (\(40\times 40\)) is applied. Figure 9: Evolution of the three unstable eigenvalue pairs using nonstationary ordinary kriging. (a) Real part; (b) Imaginary part. Next, stationary ordinary kriging is applied to create the metamodels. The response surfaces using 150 Sobol sequential samples provided by UQLab [24] are given in Figure 11. It can be seen from the results that the third eigenvalue pair is most accurate, while other pairs show oscillation behavior over the stable region. This is attributed to the fact that for the third eigenvalue pairs, the sensitivity of the bifurcation friction coefficient due to the disk's parameter is relatively low. Therefore, the response surface has nonstationary characteristics which mainly depend on the friction coefficient. However, for the first and second eigenvalue, nonstationarity exists for both dimensions. Reflecting this behavior under stationarity assumptions results in oscillatory responses. Based on the given 150 samples, our proposed method is applied. The hyperplane results of the SVM for three unstable eigenvalue pairs are shown in Figure 12. Since the Gaussian kernel is applied, the obtained hyperplanes are nonlinear, and can classify the given output classes properly. After classifying the validated samples utilizing the hyperplane, the predicted response results are presented in Figure 13. The obtained results are quite similar to the reference results in Figure 10, and no spurious oscillations occur. One observed property of the proposed method is that there is a slight peak in the second eigenvalue pair result.