With a one-mode truncation it is possible to reduce the Moore–Greitzer model for compressor instability to a set of three ordinary differential equations. These are approached from the point of view of bifurcation theory. Most of the bifurcations emerge from a degenerate Takens–Bogdanov bifurcation point. The bifurcation sets are completed using the numerical branch tracking scheme AUTO. Despite the severity of the truncation, the agreement with experimental results is excellent.