The main objective is to investigate the dynamic characteristics of the dual-rotor system, typical for most aeroengines, under the influence of inter-shaft bearing behavior and the air-induced Alford force. The finite element method (FEM) is utilized to model the dual-rotor system along with nonlinear inter-shaft bearing forces and Alford force model for air-induced forces in axial compressors. The Newmark-beta method is adopted to solve the nonlinear system equations for the dynamic responses. Alford force affects the system motion by introducing and altering the system characteristic frequencies. It intensifies the characteristic frequencies of the inter-shaft bearing. The introduction of Alford force also strengthens the coupling between the high- and low-pressure rotor and the inter-shaft bearing affecting the contact between the inter-shaft bearing roller and the raceways. With the increase of rotation speed, the influence of the inter-shaft bearing force on the system gradually decreases, while the influence of the Alford force on the system gradually increases, and the system undergoes a transition from multi-periodic motion to quasi-periodic motion and into chaos. With the increase of the rotation speed ratio (RSR) between the two rotors, the influence of the inter-shaft bearing forces on the system decreases. Moreover, the increase of the RSR will also lead to a more complex motion state of the system. The increase of the inter-shaft bearing clearance (BC) strengthens the coupling effect between the high- and low-pressure rotors while showing significant influence on the Alford force. When Alford force exists, various BC values lead to different system states as periodic, quasi-periodic and chaotic. For aeroengines, the dual-rotor system incorporating inter-shaft bearing dynamic characteristics and air-induced Alford forces undergoes complex nonlinear vibration patterns. In design, the influence of Alford forces should be comprehensively considered, especially when it interacts with the inter-shaft bearing.