## 4 Bifurcation Analysis, Stability Charts, and Control Performance
### Influence of the Proportional Gain (\(p\)) on the Rotor System Dynamics  
Before proceeding further, we should remember that the normalized system parameters are defined such that \(p=\frac{c_{0}}{\Omega}k_{1},\ d=\frac{c_{0}\Omega}{\Omega}k_{2},\) and \(f=\frac{c}{\Omega}\). Accordingly, it is easy to deduce that the dimensionless coefficients \(p\) and \(d\) represent the dimensionless form of the proportional and derivative gains, respectively, while \(f\) represents the eccentricity of the rotating disc. In addition, the detuning parameter \(\sigma\) denotes the closeness of the system angular speed \(\Omega\) to its natural frequency \(\omega\) (i.e., \(\Omega=\omega+\sigma\)). In the following, the stability conditions, bifurcation analysis, and control performance of the twelve-pole system are investigated in terms of the parameters \(p,\ d,\ f,\) and \(\sigma\). Figure 3 illustrates the angular speed response curves of the twelve-pole system at various values of the eccentricity (i.e., \(f=0.005,\ 0.01,\ 0.015\)) when the proportional gain \(p=1.22\). In general, the figure shows that the twelve-pole system has a complicated response near the perfect resonance (i.e., when \(\Omega\rightarrow\omega\)), where the rotor system may possess two and/or three stable periodic solutions besides the unstable one at the same angular speed. In addition, the figure confirms that the vibration amplitudes of the twelve-pole system are monotonic increasing function the eccentricity \(f\). Moreover, Figure 3e,f illustrate that the twelve-poles system can lose its stability at a specific angular speed interval (i.e., \(-0.002837<\sigma<0.007352\)) if \(f\) exceeds a specific critical value (i.e., \(f=0.015\)). The twelve-pole dynamics at different values of the proportional gain is explored through Figure 4, where Figure 4a,b show the system steady-state vibrations against \(\sigma\) at \(p=1.4\), while Figure 4c,d depict the system steady-state vibrations against \(\sigma\) at \(p=1.5\) when fixing the eccentricity at \(f=0.01\). Comparing Figure 3c,d with Figure 4c,d, we can notice two important phenomena. The first phenomenon (positive phenomenon) is that the system dynamical behaviors became simpler when increasing \(p=1.22\) as in Figure 3c,d to \(p=1.5\) as in Figure 4c,d, where the system exhibits a response similar to a Duffing oscillator with hard spring characteristics. The second phenomenon (negative phenomenon) is that the vibration amplitudes (\(a\) & \(b\)) at \(p=1.5\) are about twice the oscillation amplitudes when \(p=1.22\). The sensitivity of the twelve-pole system to the different magnitudes of the rotor eccentricity (i.e., \(f=0.005,\ 0.01\), and \(0.015\)) when the proportional gain kept fixed at \(p=1.5\) is illustrated in Figure 5. The figure shows that the system vibration amplitudes (\(a\) and \(b\)) are a monotonic increasing function of the eccentricity. In addition, the system behaves as a Duffing oscillator with hard spring characteristics in both the horizontal and vertical directions, where the system may have bistable periodic solutions at a specific range of angular speeds. Comparing Figures 3 and 5, it is noticed that the twelve-pole system may lose its stability to respond with aperiodic oscillations when \(\Omega\cong\omega\) (i.e., when \(\Omega\) is the same).