## 4 Bifurcation Analysis, Stability Charts, and Control Performance
### Influence of the Proportional Gain (\(p\)) on the Rotor System Dynamics
Figure 6: Oscillation amplitudes of the twelve-pole system versus \(p\) at \(\sigma=0.0,\ d=0.005,\) and \(f=0.01\): (**a**) oscillation in \(X\)-direction, (**b**) oscillation in \(Y\)-direction. Figure 7: Stability chart of the twelve-pole system: (**a**) \(p-\sigma\) plane when \(d=0.005\) and \(f=0.01,\) (**b**) \(p-f\) plane when \(d=0.005\) and \(\sigma=0.\)  
Figure 8: Numerical simulation for the twelve-pole system according to the marked point \(p_{1}\) in Figure 6a (i.e., when \(p=1.8,\sigma=-0.01\)): (**a.b**) temporal oscillations, (**c**) orbital motion, (**d.e**) Poincaré return map, and (**f.g**) frequency spectrum. Figure 9: Numerical simulation for the twelve-pole system according to the marked point \(p_{2}\) in Figure 6a (i.e., when \(p=18,\sigma=0.0\)); (**a.b**) temporal oscillations, (**c**) orbital motion, (**d.e**) Poincaré return map, and (**d.g**) frequency spectrum. Figure 5.0 Numerical simulation for the twelve-pole system according to the marked point \(p_{3}\) in Figure 6a (i.e., when \(p=2.0,\sigma=0.0\)): (**a.b**) temporal oscillations, (**c**) orbital motion, (**d.e**) Poincaré return map, and (**f.g**) frequency spectrum. Figure 5.1 Numerical simulation for the twelve-pole system according to the marked point \(p_{4}\) in Figure 6a (i.e., when \(p=2.0,\sigma=0.01\)). (**a.b**) temporal oscillation, (**c**) orbital motion, (**d.e**) Poincaré return map, and (**d.g**) frequency spectrum. Figure 9 simulates the temporal vibrations of the twelve-pole system according to the point \(p_{2}\) (i.e., \(p=1.8,\sigma=0.0\), \(f=0.01\), and \(d=0.005\)) that marked in Figure 7a within the unstable solutions region. Figure 9a,b illustrate the instantaneous oscillations of the system, while Figure 9c shows the corresponding unstable periodic orbit. Figure 9a-c show that the twelve-pole system exhibits unstable periodic oscillations when simulated according to the point \(p_{2}\) that agrees accurately with the obtained stability chart is given in Figure 7a. Moreover, to identify the nature of the system motion shown in Figure 9a,b, the corresponding Poincare-map and frequency spectrum are established as shown in Figure 9d-g. Figure 9d-g depict that the rotor system performs a quasi-periodic oscillation where the Poincare-map is obtained as a closed curve. In addition, the frequency spectrum shows that the temporal oscillations contain irrational frequency components. Figures 10 and 11 simulate the twelve-pole system oscillations according to the points \(p_{3}\) and \(p_{4}\) that are marked in Figure 7a, respectively. Figure 10 shows that the rotor system has a quasiperiodic motion when simulated according to the point \(p_{3}\) (i.e., when \(p=2.0,\sigma=0.0\), \(f=0.01\), and \(=0.005\)) that lies with the unstable solutions region. On the other hand, the system responds with stable periodic motion when simulated according to the point \(p_{4}\) (i.e., when \(p=2.0,\sigma=0.01\), \(f=0.01\), and \(=0.005\)) that lies with the stable solutions region as shown in Figure 5.2 Depending on the obtained analytical and numerical results investigations given in Figures 7-11, we can confirm the accuracy of the obtained analytical solution that given by Equations (34)-(37) in describing the steady-state motion of the twelve-pole system governed by Equations (8) and (9).