## 4 Sensitivity Investigations and Numerical Simulations
### System Dynamics with Control (\(\beta_{j}\
It is clear from Figure 14 that the Jeffcott system exhibits a quasi-periodic motion (i.e., unstable periodic solution) when \(\delta_{1}=0.85\) and \(\sigma=0.1\) on the time interval \(0\leq t\leq 700\), but when \(\delta_{1}\) has been decreased to \(0.8\), the unstable periodic solution has been turned into a stable periodic solution on interval \(700<t\leq 1000\). Comparing Figure 14 with Figure 13 at \(\sigma=0.1\), one can demonstrate the excellent correspondence between the analytical results given in Figure 13 and the numerical simulations shown in Figure 6:3  
The non-linear interactions between the Jeffcott system and the eight-pole actuator at three different values of the control parameter \(\delta_{2}\) have been illustrated in Figure 15, where Figure 15a,b show the oscillation amplitudes of the controlled Jeffcott system at \(\delta_{2}=0.005\), while Figure 15c,d depict the Jeffcott system vibrations amplitudes when \(\delta_{2}=0.01\) and \(0.02\). In general, Figure 15 shows that \(a_{1}\) and \(a_{2}\) are a monotonic decreasing function of the control parameter \(\delta_{2}\). In addition, Figure 15c,d confirm that the increase of \(\delta_{2}\) to a critical value may eliminate the complex bifurcation behavior of the rotor-actuator system to respond as a linear system regardless of the angular speed \(\Omega=1+\sigma,-0.2\leq\sigma\leq 0.2\). Based on Figures 13 and 15, we can report that the control parameters \(\delta_{1}\) and \(\delta_{2}\) can be utilized to reshape the Jeffcott system dynamics, where one can shift \(\sigma\)--response curve to the left or the right side via increasing or decreasing \(\delta_{1}\). In addition, the system oscillations amplitudes can be mitigated by increasing \(\delta_{2}\). The influence of both \(\delta_{1}\) and \(\delta_{2}\) can be explained simply from the mathematical point of view through the derived equations of motion (20) and (21) that can be rewritten as follows:  
Figure 6:4  
Figure 6:5  
Based on the above two equations, coupling a PD-control algorithm to the Jeffcott system via a magnetic actuator has added the linear terms \(\beta_{1}u,\ \beta_{1}v,\ \beta_{2}u\) and \(\beta_{2}\dot{v}\). So, the equivalent natural frequency of the controlled Jeffcott system became \(\omega^{2}_{control}=1-\beta_{1}\) and its linear damping is \(\mu_{control}=\mu-\beta_{2}\), where \(\beta_{1}=8\cos^{2}(\beta)-8\cos(\beta)\delta_{1}-4\delta_{1}+4\)and \(\beta_{2}=-4\delta_{2}(1+2\cos(\beta))\). Therefore, one can change the system's natural frequency \(\omega_{control}\) via changing \(\delta_{1}\). In addition, the system damping coefficient \(\mu_{control}\) can be controlled via the control parameter \(\delta_{2}\). Figure 15 clearly shows that the maximum oscillation amplitudes of the controlled Jeffcott system occur at the perfect resonance (i.e., when \(\Omega=1+\sigma\), \(\sigma=0\)). Accordingly, the control parameter \(\delta_{2}\) has been plotted against the vibration amplitudes (\(a_{1}\) and \(a_{2}\)) at \(\sigma=0\) to illustrate the motion bifurcation of the controlled Jeffcott system at the different values of \(\delta_{2}\) as illustrated in Figure 6:6 It is clear from the figure that the system may oscillate with one of three oscillation modes depending on \(\delta_{2}\) magnitude, where the system the Jeffcott system exhibits a bounded aperiodic (i.e., quasi-periodic, or chaotic) motion as long as on \(0\leq\delta_{2}<0.0025\). In addition, Figure 16 demonstrates that the system can oscillate with one of two stable periodic solutions according to the initial conditions (i.e., has a bi-stable periodic solution) if \(0.0025\leq\delta_{2}<0.0082\). Moreover, the controlled Jeffcott system oscillates periodically with a single periodic attractor like the linear system if \(\delta_{2}\geq 0.0082\).