Remarks on Interacting Neimark-Sacker Bifurcations

We study codimension-2 bifurcations of fixed points of dissipative diffeomorphisms with a pair of complex critical eigenvalues together with either an eigenvalue −1 or another such a pair. In the previous studies only cubic normal forms were considered. However, in some cases the unfolding requires higher order terms and these are investigated here. We (re)derive the normal forms and reduce them to a single amplitude map. This map is similar to the amplitude system for the double-Hopf bifurcation for vector fields. We show how the critical normal form coefficients determine the general bifurcation picture for this amplitude map. Generic nonsymmetric perturbations of the normal forms are considered. Our case studies show a detailed picture near various bifurcation curves, which was somewhat richer than the theoretical predictions. For arbitrary maps with these bifurcations we give explicit formulas for critical normal form coefficients on center manifolds and apply them to two examples. Here we are able to demonstrate the existence of the bubble-structure, which was only observed in unfolded normal forms before.