In order to probe into the flow mechanism in the axial-flow compressor, the nonlinear flow characteristics of the axial-flow compressor are qualitatively analyzed in this paper. Based on the multiparameter Moore-Greitzer brief model in compression systems, both stability and bifurcation of system equilibriums under bifurcation parameters can be achieved conveniently by applying the nonlinear dynamic theory. According to the analysis, the sketch maps for steady and unsteady flow of compressor are plotted in the － bifurcation parameters space in both the symmetrical and asymmetrical flow states. The results show that the nonlinear bifurcation analysis can accurately capture typical post-stall transients under bifurcation parameters. In addition, the stability analysis can be easily used for analyzing the change of stability of system equilibriums. The sketch map can be used in determining the state of axial compressor qualitatively.