H.W. Broer, M. Golubitsky and G. Vegter
The geometry of resonance tongues
In: Singularity Theory.
(D. Cheniot, N. Dutertre, C. Murolo, D. Trotman and A. Pichon, eds.)
World Scientific Publ. Co., 2007, 327-356.
Resonance tongues and tongue boundaries are studied for nondegenerate and
(certain) degenerate Hopf bifurcations of maps using singularity theory
methods of normal forms and universal unfoldings. We recover the standard
theory of tongues (the nondegenerate case) in a straightforward
way and we find certain surprises in the tongue boundary structure when
degeneracies are present. For example, the tongue boundaries at degenerate
singularities in weak resonance are much blunter than expected from the
nondegenerate theory.