BANANAS AND BANANA SPLITS - A PARAMETRIC DEGENERACY IN THE HOPF-BIFURCATION FOR MAPS

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