This paper considers the problem of maintaining a linear, discrete-time system at a desired state using a model-reference, self-adapting, feedback control scheme. In particular, we focus on the dynamics introduced by the (nonlinear) controller and observe how the behavior of the system changes as the type and magnitude of plant/reference model mismatch vary. The desired fixed point of the system loses local stability via both period-doubling and Hopf bifurcations. Starting from these local instabilities, this paper presents a computer-assisted study of the dynamic behavior of the system, and, in particular, of the resonance structures associated with the Hopf bifurcation. Special emphasis is given to undesired dynamic features, such as multiple attractors and long transients which, because of the subcritical nature of this bifurcation, coexist with a locally stable fixed point.