## 4 Bifurcation Analysis, Stability Charts, and Control Performance
### Influence of the Derivative Gain (\(d\)) on the Rotor System Dynamics  
The influence of the second control parameter (i.e., the derivative gain \(d\)) on the dynamical behaviors of the twelve-pole system is discussed through Figures 12-18. In Figure 12, the twelve-pole system steady-state oscillation amplitudes are plotted against the detuning parameter \(\sigma\) at various values of the derivative gain (i.e., when \(d=0.005\), \(0.01\), and \(0.015\)) when fixing the eccentricity and the proportional gain constant so that \(f=0.015\) and \(p=1.22\). Comparing Figure 12a,b with Figure 12c,d, it is noticed that the increase in \(d\) from \(0.005\) to \(0.01\) eliminates the unstable motions of the twelve-pole system that appeared in Figure 12a,b and reduces the vibration amplitudes (\(a\) & \(b\)) of the rotor system. However, Figure 12c,d show that the system has a bistable solution near the perfect resonance (i.e., when \(\Omega\rightarrow\omega\)), and the nonlinearity characteristics still dominate the rotor response. However, when increasing the derivative gain from \(d=0.01\) to \(d=0.015\), as depicted in Figure 12e,f, it forces the twelve-pole system to respond linearly where the bifurcations of the motions are suppressed. The effect of the derivative gain on the steady-state oscillation amplitudes of the rotor system when fixing \(p=1.5\) and \(f=0.015\) has been illustrated in Figure 13. It is clear from the figure that the oscillation amplitudes are a monotonic decreasing function of the derivative gain \(d\), where the system may respond linearly, and the motion bifurcations disappear at \(d=0.015\). Figure 12: Oscillation amplitudes of the twelve-pole system versus \(\sigma\) at \(f=0.015\) and \(p=1.22\): (**a,b**) \(d=0.005\), (**c,d**) \(d=0.01\), and (**e,f**) \(d=0.015\). Figure 14: Oscillation amplitudes of the twelve-pole system versus \(d\) when \(\sigma=0.0\) and \(f=0.01\): (**a,b**) \(p=1.22\), and (**c,d**) \(p=1.5\). Figure 13: Oscillation amplitudes of the twelve-pole system versus \(\sigma\) at \(p=1.5\), \(f=0.015\) when \(d=0.005,\ 0.01\) and \(0.015\): (**a**) oscillation in \(X\)-direction, (**b**) oscillation in \(Y\)-direction. The mechanism of the derivative gain \(d\) in eliminating the motion bifurcations and reducing the system vibration amplitudes can be explained based on the definition of the twelve-pole system linear damping coefficient \(\mu=\left(cos(\alpha)+\frac{1}{2}\right)d\). The linear damping coefficient \(\mu\) is a constant multiple of the derivative control gain \(d\). Therefore, the increase in \(d\) is a direct increase of \(\mu\), which in turn decreases the oscillation amplitudes and eliminates the bifurcation of the system motion at a specific critical value. Accordingly, the derivative gain is employed as a bifurcation parameter as shown in Figure 14, where the oscillation amplitudes are plotted against \(d\) at various values of the proportional gain \(p\) (i.e., \(p=1.22\) and \(p=1.5\)) when fixing \(f=0.01\) and \(\sigma=0.0\). Comparing Figure 14a,b with Figure 14c,d, we can deduce that the amplitudes (\(a\) & \(b\)) are a monotonic decreasing function of the derivative gain. In addition, the figures show that the system has a complex bifurcation behavior at \(p=1.22\), while increasing the proportional gain to \(p=1.5\) can eliminate the bifurcation of the system motion. Moreover, Figure 14 shows that the twelve-pole system may lose its stability if \(d<0.00435\) when fixing \(p=1.22\), while increasing \(p\) to became \(1.5\) enhances the stability margin (i.e., the system loses its stability if \(d<0.0005\) when \(p=1.5\)).