Analysis of the post-instability dynamical behavior of an axial- flow compression system model is carried out in a bifurcation- theoretic setting. Using global analysis techniques, we uncover the sequence of bifurcations in parameter space which allows us to rigorously determine whether the compressor stalls or surges when the throttle is slowly closed beyond the instability margin. Using these computational techniques, we also determine the conditions under which stalled and/or surging flow solutions coexist with the desired uniform-flow operating point and quantify the perturbations which destabilize this operating point.