## 5 Numerical Validations and Temporal Oscillations  
The bifurcation diagrams given in Figures 3-7 were obtained based on the approximate analytical solution (i.e., Equations (69)-(74)) of the controlled system equations of motion (i.e., Equations (21)-(26)), where the steady-state oscillation amplitudes (i.e., \(a_{1},\ a_{2},\ a_{3}\), and \(a_{4}\)) of Equations (69)-(74) are governed by the non-linear algebraic system given by Equation (75). Accordingly, this section is dedicated to demonstrating the accuracy of that approximate solution via solving the Equations of motion (i.e., Equations (21)-(26)) numerically using the ODE45 MATLAB solver. The numerical simulation of the steady-state vibration amplitudes (i.e., \(a_{1},\ a_{2},\ a_{3}\), and \(a_{4}\)) was illustrated as small circles (when sweeping the bifurcation parameter (\(\sigma\) or \(f\)) forward) and as big dots (when sweeping the bifurcation parameter (\(\sigma\) or \(f\)) backward) as shown in Figures 8-12. Figure 8 shows the steady-state lateral oscillations of the uncontrolled rotor system which plotted against the parameter \(\sigma\) when \(f=0.013\). In addition, the numerical validation is illustrated as small circles and big dots, where the figure demonstrates the accurate correspondence between the numerical and approximate solutions. Figure 9: Vibration amplitudes of the rotor system and PPF-controller using \(\sigma\) as a bifurcation parameter when \(f=0.013\): (**a**) vibration amplitude in \(X_{1}\) direction, (**b**) vibration amplitude in \(X_{2}\) direction, and (**c,d**) vibration amplitudes of the PPF-controller. Figure 10: Vibration amplitudes of the rotor system, LIPPF-controller, and ALIPPF-controller when \(f=0.013\): (**a**) vibration amplitude in \(X_{1}\) direction, (**b**) vibration amplitude in \(X_{2}\) direction, and (**c,d**) vibration amplitudes of the LIPPF-controller and ALIPPF-controller. Figure 11: Vibration amplitudes of the rotor system, PPF-controller, LIPPF-controller, and ALIPPF-controller using the rotor eccentricity \(f\) a bifurcation parameter when \(\sigma=0\): (**a**) vibration amplitude in \(X_{1}\) direction, (**b**) vibration amplitude in \(X_{2}\) direction, and (**c**,**d**) vibration amplitudes of the PPF-controller, LIPPF-controller, and ALIPPF-controller. The applied numerical algorithm used to validate the accuracy of the obtained approximate solution can be explained through Figure 8 as follows:  
1. Forward-sweep algorithm 1. Set \(\sigma\) as the bifurcation control parameter with an initial value \(\sigma_{initial}\ =-0.3\), step-size \(\Delta\sigma=0.02\), and final value \(\sigma_{final}\ =0.3\). 2. Set \(y_{j}(0)=0\ (j=1,2,\ldots,6)\) as zero initial conditions of the first simulation step. Figure 12: Vibration amplitudes of the rotor system, PPF-controller, LIPPF-controller, and ALIPPF-controller using the rotor eccentricity \(f\) a bifurcation parameter when \(\sigma=0.15\): **(a)** vibration amplitude in \(X_{1}\) direction, **(b)** vibration amplitude in \(X_{2}\) direction, and **(c,d)** vibration amplitudes of the PPF-controller, LIPPF-controller, and ALIPPF-controller. Set \(\sigma=\sigma_{initial}\). 4. Set \(f=0.013\), \(p=1.22\), \(d=0.005\), \(\alpha=0.3927\), \(\mu=\frac{1}{2}(2 dcos(\alpha)+d)\), \(\omega=\sqrt{2 pcos(\alpha)+p-3}\), \(\Omega=\omega+\sigma\), \(\eta_{1}=\eta_{2}=\eta_{3}=\eta_{4}=\eta_{5}=\eta_{6}=\eta_{7}=\eta_{8}=0\), \(\rho_{1}=\rho_{2}=1\), \(\mu_{1}=\mu_{2}=0.01\), \(\sigma_{1}=\sigma_{2}=0\), \(\omega_{1}=\omega+\sigma_{1}\), \(\omega_{2}=\omega+\sigma_{2}\). 5. Compute the system parameters \(\alpha_{j},\beta_{k}\), \(\gamma_{k}\) (\(j=1,\ 2,\ldots,8\), \(k=1,\ 2,\ldots,16\)) as given in the Appendix A according to the values of the parameters given in step 4. 6. Solve the system equations of motion (i.e., Equations (21)-(26)) using ODE45 MATLAB solver on the time interval \(0\leq\tau\leq 2000\) to capture the steady-state motion. 7.