###### Abstract  
The active magnetic bearings system plays a vital role in high-speed rotors technology, where many research articles have discussed the nonlinear dynamics of different categories of this system such as the four-pole, six-pole, eight-pole, and sixteen-pole systems. Although the twelve-pole system has many advantages over the eight-pole one (such as a negligible cross-coupling effect, low power consumption, better suspension behaviors, and high dynamic stiffness), the twelve-pole system oscillatory behaviors have not been studied before. Therefore, this article is assigned to explore the effect of the magneto-electro-mechanical nonlinearities on the oscillatory motion of the twelve-pole system controlled via a proportional derivative controller for the first time. The normalized equations of motion that govern the system vibrations are established by means of classical mechanics. Then, the averaging equations are extracted utilizing the asymptotic analysis. The influence of all system parameters on the steady-state oscillation amplitudes is explored. Stability charts in a two-dimensional space are constructed. The stable margin of both the system and control parameters is determined. The obtained investigations reveal that proportional gain plays a dominant role in reshaping the dynamics and motion bifurcation of the twelve-pole systems. In addition, it is found that stability charts of the system can be controlled by simply utilizing both the proportional and derivative gains. Moreover, the numerical simulations showed that the twelve-poles system can exhibit both quasiperiodic and chaotic oscillations besides the periodic motion depending on the control parameters' magnitude. type: Nonlinear Nonlinear dynamics, Stability conditions, Bifurcation analysis, Bifurcation  
rotor system within the stator in its hovering position. To make sure the provided force is controllable, the rest of the elements (i.e., the proximity sensors, digital controller, and power amplifiers) are required to form what is known as a closed-loop control system. Simply, the AMBS is works as follows: (1) the proximity sensors measure the instantaneous vibrations \(x(t)\) and \(y(t)\) of the rotor; (2) the measured signals \(x(t)\) and \(y(t)\) are sent to the digital controller to manipulate them according to the provided control algorithm; (3) the manipulated signals (i.e., control signals) are fed back again to the power amplifiers, which in turn apply controlled electrical currents to the electromagnetic poles; and (4) finally, the electromagnetic poles generate a corresponding attractive magnetic force that allows the rotor system to rotate without lateral oscillations in \(X\) and \(Y\) directions. The frictionless feature of AMBS provides many advantages over the conventional bearings system such as the clean environmental operations of the machines, less maintenance, long working time, and no necessity for lubrication. Accordingly, the huge engineering applications and many advantages of the AMBS have attracted scientists and engineers to design and investigate the dynamical characteristic of the different configurations for such systems. [PERSON] et al. [1] presented analytical investigations for the bifurcation behaviors of the four-pole AMBS. The authors concluded that the four-pole system may suffer from a sensitivity to the initial conditions. In addition, the Hopf, saddle-connection, and saddle-node bifurcations were reported. [PERSON] et al. [2; 3] studied the vibrations control of the six-pole system. They discussed the oscillatory characteristics of the system in both the radial and Cartesian control strategies. They reported that the six-pole system can exhibit small vibration amplitudes and complex bifurcation behavior in the case of Cartesian control. On the other hand, it is found that the six-pole system can exhibit a simple bifurcation behavior such as the Duffing oscillator with strong vibrations in the radial control.