Recognition of the bifurcation type of resonance in mildly degenerate Hopf-Neimark-Sacker families

This paper deals with families of diffeomorphisms, a fixed point of which undergoes a Hopf-Neimark-Sacker bifurcation with its characteristic array of resonance tongues organizing the alteration of periodic and quasi-periodic dynamics. Our interest is with the periodic dynamics as this corresponds to subharmonic periodic solutions in the case of flows. We zoom in on the shape of one such tongue, as a subset of the resonance bifurcation diagram, briefly reviewing the classical non-degenerate case, but then turning to a next case of degeneracy. It has already been established that the generic tongue geometry involves both tongues and flames. A description of this can be given in terms of contact-equivalence singularity theory, equivariant under an appropriate cyclic group given by the resonance at hand. At an intermediate stage of the theory a Lyapunov-Schmidt reduction is applied. This gives a finite classification of such bifurcation diagrams. This paper solves the ensuing recognition problem, which aims to classify any given generic family of diffeomorphisms with respect to this.