## 5 Numerical Validations and Temporal Oscillations
Find the maximum oscillation amplitudes of \(y_{j}(\tau)\) (\(j=1,2,\ldots,6\)) on the time interval \(1900\leq\tau\leq 2000\). 8. Set \(a_{j}(\sigma)=\max\{y_{j}(1900\leq\tau\leq 200)\}\), \((j=1,2,\ldots,6)\). 9. Set the initial conditions for the next simulation step such that \(y_{j}(0)=a_{j}(\sigma)\), (\(j=1,2,\ldots,6\)). 10. Increase the bifurcation parameter \(=\sigma+\Delta\sigma\). 11. If \(\sigma\leq\sigma_{final}\) go to step (4), else go to step 12. 12. Plot \(\sigma\) versus \(a_{j}(\sigma)\), (\(j=1,2,\ldots,4\)) as small circles. 13. End of forward-sweep algorithm. 2. Backward-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. 3. 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. Find the maximum oscillation amplitudes of \(y_{j}(\tau)\) (\(j=1,2,\ldots,6\)) on the time interval \(1900\leq\tau\leq 2000\). 8. Set \(a_{j}(\sigma)=\max\{y_{j}(1900\leq\tau\leq 200)\}\), (\(j=1,2,\ldots,6\)). 9. Set the initial conditions for the next simulation step such that \(y_{j}(0)=a_{j}(\sigma)\), (\(j=1,2,\ldots,6\)). 10. Increase the bifurcation parameter \(=\sigma+\Delta\sigma\). 11. If \(\sigma\geq\sigma_{final}\) go to step (4), else go to step 12. 12. Plot \(\sigma\) versus \(a_{j}(\sigma)\), (\(j=1,2,\ldots,4\)) as big dots. 13. End of backward-sweep algorithm  
According to the above algorithms, the accuracy of the obtained bifurcation diagrams (obtained in the three-dimensional space as in Section 4) was validated numerically within this section as shown in Figures 9-12, where Figure 9 shows the steady-state oscillation amplitudes of both the rotor system and the conventional PPF-controller only (i.e., when \(\eta_{1}=\eta_{3}=\eta_{5}=\eta_{6}=0.2\), \(\eta_{2}=\eta_{4}=\eta_{7}=\eta_{8}=0\)) when \(f=0.013\). The figure illustrates that the connection of the PPF-controller to the rotor system can eliminate the system vibrations in the vicinity of \(\sigma=0\). However, the analytical and numerical solutions confirm that the considered system may exhibit high oscillation amplitudes on the left and right sides of \(\sigma=0\). On the other hand, Figure 10 shows the oscillatory behavior of the rotor system along the \(\sigma\)-axis at \(f=0.013\) when the LIPPF-controller or the ALIPPF-controller are coupled to the rotor system, independently. It is clear from the figure that the coupling of the LIPPF-controller (i.e., when \(\eta_{1}=\eta_{3}=\eta_{5}=\eta_{6}=0.2\), \(\eta_{2}\eta_{7}=\eta_{4}\eta_{8}=0.2\)) to the rotor system eliminated the lateral vibrations in the vicinity of \(\sigma=0.0\) and suppressed the oscillation amplitudes on both sides of \(\sigma=0\). In addition, the numerical and analytical investigations demonstrated that the coupling of the ALIPPF-controller (i.e., when \(\eta_{1}=\eta_{3}=\eta_{5}=\eta_{6}=0.2\), \(\eta_{2}\eta_{7}=\eta_{4}\eta_{8}=0.2\), \(\sigma_{1}=\sigma_{2}=\sigma\)) to the rotor system eliminated the oscillation amplitude regardless of the angular velocity value.