## 3 Analytical Investigation and Autonomous Amplitude-Phase Equations
\begin{array}{ll}\sigma+\frac{1}{8}(\frac{3\alpha_{1}}{\omega}+\alpha_{6} \omega)b^{2}+\frac{1}{4}(\frac{\alpha_{2}}{\omega}+\alpha_{5}\omega)a^{2}+ \frac{1}{8}(\frac{\alpha_{2}}{\omega}-\alpha_{5}\omega)a^{2}\cos(2\varphi_{2}- 2\varphi_{1})-\frac{1}{8}(-\alpha_{4}+\alpha_{7})a^{2}\\ \times\sin(2\varphi_{2}-2\varphi_{1})+\frac{1}{2\omega}(\omega+\sigma)^{2}f \sin\varphi_{2}=0\\ \end{array}  \tag{33}\]  
The above nonlinear algebraic system governs both the steady-state vibration amplitudes (\(a\) and \(b\)) and the phase angles (\(\phi_{1}\) and \(\phi_{2}\)) of the twelve-pole system in terms of the rotor eccentricity (\(f\)), the angular speed (\(\sigma=\Omega-\omega\), and the control gains (\(p\) and \(d\)). So, by solving Equations (34)-(37) simultaneously using one of the petameters (i.e., \(\sigma\), \(f\), \(p\), \(d\)) as a bifurcation parameter, we can obtain the different response curves as in Figures 3 and 4, etc. In addition, the solution stability of Equations (34)-(37) can be checked via using the Jacobian matrix of the corresponding dynamical system given by Equations (28)-(31). To check the solution stability of Equations (34)-(37), let the solution of Equations (34)-(37) is (\(a_{0},\ b_{0},\varphi_{10},\varphi_{20}\)) and assume (\(a_{1},\ b_{1},\varphi_{11},\varphi_{21}\)) is a small deviation from this solution. Accordingly, we can write  
\[a=a_{0}+a_{1},\ b=b_{0}+b_{1},\ \varphi_{1}=\varphi_{10}+\varphi_{11},\ \varphi_{2}=\varphi_{20}+\varphi_{21} \tag{34}\]Figure 3: Oscillation amplitudes of the twelve-pole system versus \(\sigma\) at \(p=1.22\): (**a,b**) \(f=0.005\), (**c,d**) \(f=0.01\), and (**e,f**) \(f=0.015\). By differentiating Equation (38), we have  
\[\frac{da}{dt}=\frac{da_{1}}{dt},\;\frac{db}{dt}=\frac{db_{1}}{dt},\;\frac{d\varphi _{1}}{dt}=\frac{d\varphi_{11}}{dt},\;\frac{d\varphi_{2}}{dt}=\frac{d\varphi_{21 }}{dt} \tag{39}\]  
Substituting Equations (38) and (39) into Equations (28)-(31) and expanding for the small deviations \(a_{1},\;b_{1},\varphi_{11},\) and \(\varphi_{21}\) keeping the linear terms only, we have  
\[\left( \begin{array}{c}\frac{da_{1}}{dt}\\ \frac{db_{1}}{dt}\\ \frac{d\varphi_{11}}{dt}\\ \frac{d\varphi_{21}}{dt}\\ \end{array} \right)=\left( \begin{array}{cccc}\frac{\partial F_{1}}{\partial a _{1}}&\frac{\partial F_{1}}{\partial b_{1}}&\frac{\partial F_{1}}{\partial a_{ 1}}&\frac{\partial F_{1}}{\partial a_{21}}\\ \frac{\partial F_{2}}{\partial a_{2}}&\frac{\partial F_{2}}{\partial b_{2}}& \frac{\partial F_{2}}{\partial a_{2}}\\ \frac{\partial F_{2}}{\partial a_{2}}&\frac{\partial F_{2}}{\partial b_{2}}& \frac{\partial F_{2}}{\partial a_{2}}\\ \end{array} \right)\left( \begin{array}{c}a_{1}\\ b_{1}\\ \varphi_{11}\\ \varphi_{21}\\ \end{array} \right) \tag{40}\]  
The coefficients of the Jacobian matrix are given below in Appendix C. According to the [PERSON] theorem, the linear system given by Equation (40) is topologically equivalent to the nonlinear system given by Equations (28)-(31). Therefore, the solution stability of Equations (34)-(37) can be explored via the eigenvalues of the Jacobian matrix. Figure 4: Oscillation amplitudes of the twelve-pole system versus \(\sigma\) at \(f=0.01\): (**a,b**) \(p=1.4\), and (**c,d**) \(p=1.5\).