Initial geometric imperfections arising from manufacturing processes and service conditions serve as a critical trigger for dynamic instability of structures. However, the influence pathways of such imperfections on structural dynamic behavior remain unclear. In particular, when it comes to complex nonlinear responses such as bifurcation and chaos, the coupled instability mechanisms involving initial geometric imperfections, structural motion states, and complex physical environments remain largely unexplored. To address this gap, a rotating functionally graded (FGM) cylindrical shell with initial geometric imperfections is adopted as the subject of investigation. The bifurcation and chaotic responses of the shell are systematically examined under coupled multi-physical fields, including thermal environments, magnetic fields, and external excitations. First, a magneto-thermo-elastic nonlinear dynamic model is established via Hamilton's principle, which comprehensively accounts for geometric nonlinearity, thermoelastic nonlinearity, and magnetization nonlinearity. The proposed model captures the effects of initial geometric imperfections, rotational motion, and spatially graded material properties. Subsequently, the partial differential governing equations are discretized into ordinary differential equations via the Galerkin method, and the critical condition for the onset of Smale horseshoe chaos is analytically derived using the Melnikov method. Finally, a parametric study on the bifurcation and chaotic behaviors is conducted through numerical examples, with the aim of elucidating the influence mechanisms of various control parameters on the system's bifurcation and chaotic responses. The results indicate that initial geometric imperfections alter the sensitivity of the dynamic system to external multi-field environments by modulating the nonlinear characteristics of the structure. Meanwhile, the magnetic field intensity exerts a suppressive effect on chaos through electromagnetic damping, whereas the external excitation promotes the occurrence of chaos via energy injection.