Henk W Broer et al 2003 Nonlinearity 16 1511 doi:10.1088/0951-7715/16/4/319
The geometry of resonance tongues: a singularity theory approach
Henk W Broer1, Martin Golubitsky2 and Gert Vegter3
Show affiliationsRecommended by A Chenciner
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.
- PACS
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05.45.-a Nonlinear dynamics and nonlinear dynamical systems
- MSC
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37G15 Bifurcations of limit cycles and periodic orbits
- Subjects
- Dates
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Issue 4 (July 2003)
Received 7 November 2002, in final form 20 May 2003
Published 6 June 2003
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The geometry of resonance tongues: a singularity theory approach
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