## 4 Bifurcation Analysis, Stability Charts, and Control Performance
### Influence of the Proportional Gain (\(p\)) on the Rotor System Dynamics
\(\sigma=0.0\)) if the eccentricity \(f=0.015\) and the proportional gain \(p=1.22\) (see Figure 3e,f), meanwhile increasing the proportional gain from \(p=1.22\) to \(p=1.5\), eliminates the instability conditions as shown in Figure 5.  
Due to the dominant role of the proportional gain (\(p\)) in reshaping the vibration characteristics of the twelve-pole system as demonstrated in Figures 3-5, the proportional gain has been employed as a bifurcation parameter where \(p\) is plotted versus the vibration amplitudes (\(a\) & \(b\)) of the twelve-pole system when \(f=0.01\) and \(\sigma=0.0\) as shown in Figure 6. Clear from the figure is the great influence of the proportional gain on the oscillation amplitudes and bifurcation of the system motion, where the figure shows that the twelve-pole system has no solution as long as the proportional gain \(p<1.098\). In addition, the figure depicts that the periodic solution of the twelve-pole system may lose its stability if the gain \(p\) exceeds a specific limit (i.e., when \(p>1.778\)). Moreover, Figure 6 demonstrates that the system may respond with a unique periodic solution or bistable periodic solutions depending on the \(p\) magnitude within the interval \(1.098<p<1.778\). It is simple to explain the absence of the solution for the twelve-pole system from the engineering point of view when \(p<1.098\). By coming back to the definition of the system natural frequency \(\omega=\sqrt{2 pos(\alpha)+p-3}\), where \(\omega\) depends only on the proportional gain because \(\alpha\) is constant (i.e., \(\alpha=30^{\circ}\) see Figure 3). Accordingly, the selection of the proportional gain in a way that makes \(\omega^{2}<0\) (i.e., \(p\leq 1.098\)) is not acceptable from the engineering standpoint, where the system has no solution if its natural frequency is an imaginary value. To obtain the stability boundaries of proportional control gain in more general form, the stability charts for the twelve-pole system are obtained in both \(p-\sigma\), and \(p-f\) planes, as in Figure 7. The figure illustrates that the system has no solution as long \(p<1.098\) regardless of the values of \(\sigma\) and \(f\), as demonstrated in Figure 7a,b, respectively, where the absence of the solution depends on the magnitude of \(p\) and hence the magnitude of the natural frequency \(\omega\) as explained above. To investigate the accuracy of the obtained stability charts shown in Figure 7, a numerical simulation for the twelve-pole system (i.e., solving Equations (8) and (9) using MATLAB ODE45) according to four points \(p_{1},\ p_{2},p_{3}\), and \(p_{4}\) that are shown in Figure 7a and are introduced in Figure 8, Figure 9, Figure 10, Figure 11. It is clear that the point \(p_{1}\) is located within the stable solution region. Accordingly, it is expected that the twelve-pole rotor to respond with stable periodic oscillation when tuning the system parameters such that \(p=1.8,\sigma=-0.01,\ f=0.01\), and \(d=0.005\). Figure 8 shows the temporal dynamics of the twelve-pole system according to the point \(p_{1}\) (i.e., \(p=1.8,\sigma=-0.01,\ f=0.01\),  
Figure 5: Oscillation amplitudes of the twelve-pole system versus \(\sigma\) at \(p=1.5\) when \(f=0.005\), \(0.01\) and \(0.015\): (**a**) oscillation amplitude in \(X\)-direction, (**b**) oscillation amplitude in \(Y\)-direction. and \(d=0.005\)), where Figure 8a,b show the instantaneous oscillations of the system, and Figure 8c illustrates the steady-state periodic orbit. In addition, both the corresponding Poincare-map and frequency spectrum for the system temporal oscillations is constructed, as shown in Figure 8d-g. In general, Figure 8 confirms the stable periodic motion for the studied system at \(p=1.8,\sigma=-0.01,\)\(f=0.01,\) and \(d=0.005,\) where the orbital motion, the Poincare-map, and the frequency spectrum demonstrate that the temporal oscillations contain a single frequency component.