## 4 Bifurcation Analysis and Control Performance  
The performance of the introduced LIPPF-controller in eliminating the rotor vibration amplitudes (\(a_{1},\ a_{2}\)) is discussed within this section via solving Equation (75) numerically at different values of the control gains (\(\eta_{1},\eta_{2},\eta_{3},\eta_{4}\)) and feedback gains (\(\eta_{5},\eta_{6},\eta_{7},\eta_{8}\)) utilizing \(\sigma\) or \(f\) as the bifurcation parameter [49]. In addition, the solution stability of Equation (75) was explored according to the eigenvalues of the derived variational equations (i.e., Equations (77)-(84)), where the stable solution is plotted as a solid line and the unstable solution is illustrated as a red dotted line. To simulate the steady-state and transient oscillations of the studied controlled system, the parameters' values were adopted as follows unless otherwise mentioned [29; 39]: \(p=1.22\), \(d=0.005\), \(\alpha=22.5^{\circ}\), \(\mu_{1}=\mu_{2}=0.01\), \(\eta_{1}=\eta_{2}=\eta_{3}=\eta_{4}=\eta_{5}=\eta_{6}=0.2\), \(\eta_{7}=\eta_{8}=\rho_{1}=\rho_{2}=1\), \(\Omega=\omega+\sigma\), \(\omega_{1}=\omega+\sigma_{1}\), \(\omega_{2}=\omega+\sigma_{2}\), \(\sigma_{1}=\sigma_{2}=0\), and \(f=0.015\). Before proceeding further, it is important to remember that the parameter \(\sigma\) is employed to represent the closeness of the angular speed (\(\Omega\)) of the rotor system to its natural frequency \(\omega=\sqrt{2 pcos(\alpha)+p-3}\) as given in Equation (55). Accordingly, in the whole article \(\sigma\) is used as a bifurcation parameter to explore the nature of the rotor vibrations at the primary resonance case (i.e., when \(\Omega\) is approaching \(\omega\) from the right or the left side). The rotor oscillation amplitudes (\(a_{1}\) and \(a_{2}\)) were obtained as shown in Figure 3 via solving Equation (75) using \(\sigma\) as the main bifurcation parameter at various levels of the rotor eccentricity (\(f\)) before activating the LIPPF-controller (i.e., when \(\eta_{j}=0,\ j=1,2,\ldots,8\)). It is clear from the figure that the uncontrolled system can behave as a linear dynamical system at the primary resonance case (i.e., \(\sigma\to 0\)) when the rotor eccentricity is too small (i.e., \(f=0.001\)). However, the figure demonstrates that the increase in the rotor eccentricity beyond \(0.001\) is resulting in high oscillation amplitudes and complex oscillatory motion of the considered system. In addition, the figure illustrates that the uncontrolled system may have two or three stable periodic solutions in the vicinity of \(\sigma=0\) depending on the eccentricity magnitude. Moreover, the system may lose its stability when the rotor eccentricity \(f\) becomes \(0.013\). Based on the introduced dimensionless variables (i.e., \(y_{1}=\frac{x_{1}}{y_{0}}\), and \(y_{2}=\frac{x_{2}}{y_{0}}\)) given before Equation (21), one can deduce that \(y_{1}\) and \(y_{2}\) represent the relative oscillations of the rotor system in \(X_{1}\) and \(X_{2}\) directions with respect to the air-gap size (\(s_{0}\)). In addition, it was illustrated that \(y_{1}(\tau)\) and \(y_{2}(\tau)\) are periodic functions as given in Equations (69) and (70). Accordingly, the actual instantaneous vibrational motion of the rotor in \(X_{1}\) and \(X_{2}\) directions can be expressed such that \(x_{1}(t)=s_{0}a_{1}(t)\cos(\Omega t-\varphi_{1}(t))\) and \(x_{2}(t)=s_{0}a_{2}(t)\cos(\Omega t-\varphi_{1}(t))\), where \(s_{0}a_{1}\) and \(s_{0}a_{2}\) are the steady-state vibration amplitudes in \(X_{1}\) and \(X_{2}\) directions, respectively. Moreover, these oscillation amplitudes should be less than the air-gap size (i.e., \(s_{0}a_{1}<s_{0}\) and \(s_{0}a_{2}<s_{0}\)) to avoid the catastrophic rub-impact force between the rotor and the magnetic pole housing. This means that the system can oscillate without rub-impact occurrence as long as \(a_{1}<1\) and \(a_{2}<1\).