IOP PUBLISHING NONLINEARITY Nonlinearity 21 (2008) 2463–2482 doi:10.1088/0951-7715/21/10/014 Recognition of the bifurcation type of resonance in mildly degenerate Hopf–Ne˘ ımark–Sacker families H W Broer, S J Holtman and G Vegter Institute for Mathematics and Computer Science, University of Groningen, The Netherlands E-mail: H.W.Broer@math.rug.nl, S.J.Holtman@math.rug.nl and G.Vegter@math.rug.nl Received 10 April 2008, in final form 6 August 2008 Published 16 September 2008 Online at stacks.iop.org/Non/21/2463 Recommended by A Chenciner Abstract This paper deals with families of diffeomorphisms, a fixed point of which undergoes a Hopf–Ne˘ ımark–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. Mathematics Subject Classification: 37G15, 37G40, 34C25 (Some figures in this article are in colour only in the electronic version) 1. Introduction Resonance tongues. We continue our study [2] of resonance tongues and their boundaries for non-degenerate and (certain) degenerate Hopf bifurcations of maps. Such bifurcations occur if one of the maps—the central singularity—in such a family exhibits resonant dynamics near a fixed point where the eigenvalues cross the unit circle. This study zooms in on the case where this occurs at a q th root of unity. Resonance tongues are regions in parameter space 0951-7715/08/102463+20$30.00 © 2008 IOP Publishing Ltd and London Mathematical Society Printed in the UK 2463