Global stable region and stability of nonlinear axial compressor system are theoretically qualitatively analyzed in this paper, Criterion of stability of the system are presented by parameters . Based on the MG model of the nonlinear axial flow compressor systems and by applying the nonlinear dynamical theory, qualitative dynamical characters are analyzed for the limited singular point firstly. The stable conditions are achieved on the basis of the various matching of parameters . Simulation results prove that when [WTBX]γ=1.2,ψc0=1.5,β=0.67[WTBZ] singular point is center point; when γ=1.4,ψc0=1.75,β=1singular point is focus;when γ=1.2,ψc0=2.2,β=2singular point is node. Secondly, qualitative dynamical characters are analyzed for the unlimited singular point, the global stable region can be divided by the nonlinear axial flow compressor systems. In addition, the relation are researched between the stable conditions,global stable region and rotating stalls as well as surge. It has academic significance on achieving reliable active control of the nonlinear axial flow compressor and understanding the relations between the system's behaviors and the parameters.