The flow in a completely filled cylinder driven by a rotating endwall has multiple time-dependent stable states when the endwall rotation exceeds a critical value. These states have been observed experimentally and computed numerically elsewhere. In this article, the linear stability of the basic state, which is a non-trivial axisymmetric flow, is analysed at parameter values where the unsteady solutions exist. We show that the basic state undergoes a succession of Hopf bifurcations and the corresponding eigenvalues and eigenvectors of these excited modes describe most of the characteristics of the observed time-dependent states.