The geometry of resonance tongues: a singularity theory approach

Abstract

Resonance tongues and their boundaries are studied for nondegenerate and (certain) degenerate Hopf bifurcations of maps using singularity theory methods of equivariant contact equivalence 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. Also at a semi-global level we find 'pockets' or 'flames' that can be understood in terms of the swallowtail catastrophe.

Resonance and Fractal Geometry

“…− The above results, summarized from [26,34,38,43], are mainly obtained by Z qequivariant Singularity Theory.…”

“…Generally on the covering space equivariant Singularity Theory can be practised, see [26,56,57] and references therein, as well as equivariant KAM Theory [23,27,33,37,44]. In the present case this construction is only needed for p/q = k/2.…”

“…We consider a fixed resonance (ν 1 , ν 2 ) = (0, p 2 /q 2 ) with p and q relatively Fig. 11 Tongue in nondegenerate resonance q = 7 of the Hopf-Neȋmark-Sacker bifurcation [26,34,38,43] prime, where we will need 2 parameters to versally unfold the linear part [3,44]. Our main interest is with the periodic points of period q, so in solving the equation…”

“…The way to study this discriminant set is by Z q -equivariant contact equivalence [26,34,38,40,43]. In the present non-degenerate case (22) can be reduced to the polynomial normal form…”

“…Two-dimensional tomography in a mildly degenerate resonance q = 7 of the Hopf-Neȋmark-Sacker bifurcation [26,34,38,43] − The strong resonances with q = 1, 2, 3 and 4 form a completely different story where the Singularity Theory is far more involved [3,86]. Still, since the higher order resonances accumulate at the boundaries, there is fractal geometry around, always of positive measure.…”