The set of Hopf bifurcations for a two-parameter family of maps is typically a curve in the parameter plane. The side of the curve on which the invariant circle exists is further divided by horn-shaped resonance regions, with each region corresponding to maps that have a periodic orbit of a certain period. With the presence of a parametric degeneracy, the resonance regions sometimes take the form of closed ''bananas'' instead of open-ended horns. The authors investigate this local codimension-two bifurcation, emphasizing resonance regions as projections to the parameter plane of surfaces in phase x parameter space. The authors present scenarios where the degeneracy occurs ''naturally'' and illustrate them through an adaptive control application. More global implications of the local study are also discussed.