## 4 Sensitivity Investigations and Numerical Simulations
### System Dynamics with Control (\(\beta_{j}\
According to Figure 7, the temporal oscillations of the controlled Jeffcott system have been illustrated when \(E=0.02,\ 0.05\), and \(0.075\) as shown in Figures 8-10, respectively, via solving the temporal Equations (20) and (21) at the zero initial conditions (i.e., \(u(0)=\dot{u}(0)=v(0)=\dot{v}(0)=0\)). Figure 8 shows the temporal oscillations (i.e., Figure 8a,b), Poincare-map (i.e., Figure 8c,d), and frequency-spectrum (i.e., Figure 8e,f, of the system when \(E=0.02\). The figure demonstrates that the Jeffcott system can perform stable periodic motions when \(E=0.02\), but these motions are not symmetric in both \(X\) and \(Y\) directions that agree with Figure 7 accurately. Figures 9 and 10 are a repetition of Figure 8, but at \(E=0.05\) and \(E=0.075\), respectively. It is clear from Figure 9 that the controlled system performs asymmetric quasi-periodic motions at \(E=0.05\), while Figure 10 confirms that the controlled Jeffcott system exhibits a chaotic response when \(E=0.075\). Figure 6: \(\sigma\)—response curve of the controlled Jeffcott rotor (i.e., \(\delta_{1}=0.83\), and \(\delta_{2}=0.002\)) at three different values of the shaft eccentricity \(E\): (**a,b**) oscillation amplitudes (\(a_{1}\) and \(a_{2}\)) versus \(\sigma\) when \(E=0.01\), (**c,d**) oscillation amplitudes (\(a_{1}\) and \(a_{2}\)) versus \(\sigma\) when \(E=0.02\), and (**e,f**) oscillation amplitudes (\(a_{1}\) and \(a_{2}\)) versus \(\sigma\) when \(E=0.03\). Figure 8: _Cont_. Figure 7: _E_—response curve of the controlled Jeffcott rotor (i.e., \(\delta_{1}=0.83\) and \(\delta_{2}=0.002\)) and the corresponding bifurcation diagrams at \(\sigma=0.0\): (**a**,**b**) oscillation amplitudes (\(a_{1}\) and \(a_{2}\)) versus \(E\) when \(\sigma=0\), and (**c**,**d**) Poincaré-map of the instantaneous oscillations \(u(t)\) and \(v(t)\) versus \(E\) when \(\sigma=0\). Based on the soft-spring characteristic of the controlled Jeffcott-system reported in Figure 6, the eccentricity response curve has been visualized when \(\sigma=-0.05\) as shown in Figure 11a,b. The figure illustrates a complex bifurcation behavior where the system may perform mono-stable, bi-stable, or tri-stable periodic motion depending on the eccentricity magnitude. In addition, the figure demonstrates that the Jeffcott system may lose its stability to oscillate with bounded aperiodic motion when \(0.063<E\leq 0.075\). Based on response curves given in Figure 11a,b, the corresponding bifurcation diagrams have been established as shown in Figure 11c,d via obtaining the Poincare-map for Equations (20) and  
Figure 8: Periodic time-response, Poincaré-map, and frequency-spectrum of the controlled Jeffcott-rotor according to Figure 7 (i.e., at \(\sigma=0,\ \delta_{1}=0.83,\ \delta_{2}=0.002\)) when \(E=0.02\): (**a,b**) the normalized temporal oscillations \(u(t)\) and \(v(t)\) in \(X\) and \(Y\) direction, (**c,d**) the corresponding Poincaré-map, (**e,f**) the corresponding frequency-spectrum. Figure 9: Quasi-periodic time-response, Poincaré-map, and frequency-spectrum of the controlled Jeffcott-rotor according to Figure 7 (i.e., at \(\sigma=0,\ \delta_{1}=0.83,\ \delta_{2}=0.002\)) when \(E=0.05\): (**a,b**) the normalized temporal oscillations \(u(t)\) and \(v(t)\) in \(X\) and \(Y\) direction, (**c,d**) the corresponding Poincaré-map, (**e,f**) the corresponding frequency-spectrum. (21) when \(\Omega=1-0.05\). By comparing Figure 11a,b with Figure 11c,d, one can note the excellent correspondence between the analytical investigations and the numerical results. Figure 11: Cont. Figure 10: Chaotic time-response, Poincaré-map, and frequency-spectrum of the controlled Jeffcott-rotor according to Figure 7 (i.e., at \(\sigma=0\), \(\delta_{1}=0.83\), \(\delta_{2}=0.002\)) when \(E=0.075\): (**a,b**) the normalized temporal oscillations \(u(t)\) and \(v(t)\) in \(X\) and \(Y\) direction, (**c,d**) the corresponding Poincaré-map, (**e,f**) the corresponding frequency-spectrum.