## 4 Bifurcation Analysis, Stability Charts, and Control Performance
### Influence of the Derivative Gain (\(d\)) on the Rotor System Dynamics
However, the same Figures (Figure 19a,b) illustrate that the rotor system has a unique stable solution along the \(f\) axis when fixing the proportional control gain to be \(p=1.5\). Accordingly, we can conclude that the twelve-pole system is robust against instability when subjected to large eccentricities when setting \(p=1.5\). Depending on Figure 19a,b, the system bifurcation diagrams are established, as shown in Figure 19c,d, utilizing \(f\) as the bifurcation parameter when \(p=1.22\) and \(p=1.5\). The bifurcation diagrams in Figure 19c,d are obtained via plotting the Poincare-map of Equations (8) and (9) versus the eccentricity \(f\) with a step-size \(\Delta f=10^{-4}\) and zero initial conditions. Comparing Figure 19a,b with Figure 19c,d, we can confirm the excellent corresponding between the analytic (i.e., Figure 19a,b) and numerical (i.e., Figure 19c,d) results. Figures 20 and 21 are a repetition for Figure 19, but when \(\sigma=-0.025\) and \(\sigma=0.025\), respectively. By examining Figures 19-21 carefully, we can notice three important phenomena. The first one is that the detuning parameter \(\sigma\) has a great influence on the critical value of the eccentricity at which the system may lose its stability when \(p=1.22\). The second phenomenon is that the twelve-pole system is stable along the \(f\) axis regardless of the magnitudes of \(\sigma\) if the proportional gain \(p=1.5\). The third phenomenon is that the steady-state oscillation amplitudes when \(p=1.5\) are greater than the steady-state oscillation amplitudes at \(p=1.22\) along the stable interval of the \(f\) axis. Figure 19: _Cont._  
Figure 12:0 **(a,b)** Oscillation amplitudes of the twelve-pole system versus \(f\), and (**c,d**) the corresponding bifurcation diagrams at \(\sigma=0.0,\ d=0.005\) when \(p=1.22\) and \(1.5\). Figure 12:1 **(a,b)** Oscillation amplitudes of the twelve-pole system versus \(f\), and (**c,d**) the corresponding bifurcation diagrams at \(\sigma=-0.025,\ d=0.005\) when \(p=1.22\) and \(1.5\). As the detuning parameter \(\sigma\) has a great influence on the critical value of the eccentricity at which the system may lose its stability as depicted in Figures 19-21, the stability chart of the twelve-pole system has been constructed in the \(f-\sigma\) plane, as shown in Figure 22a,b, when \(p=1.22\) and \(p=1.5\), respectively. It is clear from Figure 22a that the twelve-pole system may lose its stability at small eccentricity values when \(\sigma=0.0\). However, increasing or decreasing \(\sigma\) beyond zero increases the stability margin of the system. On the other hand, the stability chart in Figure 22b depicted that the system is stable for all values of the eccentricity regardless of the detuning parameter magnitude when setting \(p=1.5\). Comparing Figures 19-21 with Figure 22, we can deduce that Figure 22 is a full picture for Figures 19-21 regarding the stability boundaries of \(f\) at the different values of \(\sigma\) either when \(p=1.22\) or \(p=1.5\). Figure 12:2 (**a,b**) Oscillation amplitudes of the twelve-pole system versus \(f\), and (**c,d**) the corresponding bifurcation diagrams at \(\sigma=0.025,\ d=0.005\) when \(p=1.22\) and \(1.5\). Numerical confirmation for the stability charts shown in Figure 22, according to the marked points \(p_{1}\) and \(p_{2}\), is illustrated in Figures 23 and 24 by solving Equations (8) and (9) using MATLAB ode45. Figure 23 illustrates the twelve-pole system temporal oscillations according to the point \(p_{1}\) (i.e., \(f=0.05\), \(\sigma=0.0\), \(p=1.22\), \(d=0.005\)) that is marked in Figure 22a within the unstable solutions region. Figure 23d-g depict that the twelve-pole rotor performs chaotic motion as the Poincare map and frequency spectrum illustrated, which agrees with the stability chart in Figure 22a.