###### Abstract
The eight-pole AMBS with different configurations and control techniques has been extensively investigated [4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17]. [PERSON] et al. [4; 5] and [PERSON] et al. [6] presented theoretical and numerical investigations for the eight-pole system at the primary and super harmonic resonance conditions. The authors reported the dominance of the nonlinearities and the complexity of the motion bifurcation. The modal oscillations of the eight-pole AMBS was examined by [PERSON] et al. [7]. The system steady-state orbital oscillations were investigated utilizing the asymptotic analysis along with the phase-difference energy-ratio method. The introduced analysis illustrated that the system could perform either quasiperiodic or elliptic motion only. [PERSON] et al. [8; 9] introduced two novel control methods to mitigate the nonlinear vibrations of the eight-pole system at the primary resonance. The first control technique was to couple the rotor system to a second-order filter linearly, while the second control method was to connect the rotor system to a second-order filter in a nonlinear form. Recently, [PERSON] et al. [10] investigated the influence of the dynamical behaviors of the eight-pole AMBS including the rub-impact force between the rotor and stator. The authors concluded that the eight-pole system may respond with periodic, quasiperiodic, or chaotic motion depending on the magnitude of the impact stiffness coefficient and the value of dynamical friction. The eight-pole rotor system with time varying stiffness coefficient was investigated extensively by [PERSON] et al. [11; 12; 13; 14; 15; 16], where Shilnikov multi-pulse chaotic behavior was reported. The dynamical characteristics and motion bifurcation of the sixteen-pole AMBS system were explored under different control algorithms [17; 18; 19; 20; 21; 22; 23]. [PERSON] et al. [17; 18] introduced a detailed investigation for the sixteen-pole system with constant stiffness coefficients. They studied the influence of Cartesian and radial control methodologies on both the bifurcation characteristics and vibration reduction of the AMBS system. The authors reported that the rotor system has quartic stable periodic motions in the case of Cartesian control, while the system behaves like a Duffing oscillator with a hardening spring characteristic at the radial control case. [PERSON] and his team introduced a detailed investigation for the sixteen-pole AMBS' having time-varying stiffness coefficients [19; 20; 21; 22; 23]. The capability of the active magnetic bearings system on generating a controllable magnetic attractive force without physical contact with the rotor has attracted scientists and engineers to employ the AMBS with some control algorithms to serve as an adaptive actuator that can be used to reshape the dynamical behaviors of some rotating machinery [24; 25; 26; 27; 28; 29; 30; 31; 32; 33]. [PERSON] and [PERSON] [24] proposed a novel nonlinear absorber using four-pole AMBS to suppress the lateral vibration of a nonlinear rotor system. [PERSON] et al. [25; 26; 27; 28; 29; 30; 31; 32] employed the AMBS as an active actuator to control and reshape the oscillatory behaviors of the rotating machines by applying different control techniques. [PERSON] et al. [33] introduced an important review study that summarized the huge applications of the rotor active magnetic bearings system. According to the above discussions, it is noticed that nonlinear dynamical behaviors of the twelve-pole rotor system have not been studied before [34; 35; 36; 37]. However, the twelve-pole system has many advantages over the eight-pole system such as low power consumption, negligible cross-coupling, better suspension behaviors, and high dynamic stiffness [37]. Therefore, this article is dedicated to exploring the dynamical characteristics of the twelve-pole system for the first time. The nonlinear equations of motion that govern the system dynamics in \(X\) and \(Y\) directions are established, and the corresponding averaging equations are extracted.