This study presents a comprehensive analysis of aeroelastic stability in multi-stage turbine rotors mounted on nonlinear supports. A high-fidelity dynamic model is developed by coupling the structural behavior of a rotating shaft–disk–blade assembly with quasi-steady aerodynamic forces. The system incorporates nonlinear stiffness and damping in the bearing supports, and the governing equations of motion are derived using the Lagrangian method. Aerodynamic forces are modeled using cascade theory for incompressible subsonic flow and integrated with structural dynamics through coordinate transformation. The resulting nonlinear system is solved using the Runge-Kutta method, and its stability characteristics are investigated via bifurcation diagrams and Poincaré maps. A detailed parametric study is conducted to examine the influence of aerodynamic parameters, structural parameters and support characteristics on rotor response. Results show that nonlinear supports significantly alter stability boundaries, reduce critical flutter speeds, and introduce multi-periodic dynamic behavior. These findings provide valuable insights into the design and tuning of support systems to enhance the dynamic robustness of turbomachinery.