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Generic twistless bifurcations

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posted on 16.02.2006 by Holger R. Dullin, J.D. Meiss, D.G. Sterling
We show that in the neighborhood of the tripling bifurcation of a periodic orbit of a Hamiltonian flowor of a fixed point of an areapreserving map, there is generically a bifurcation that creates a “twistless” torus. At this bifurcation, the twist, which is the derivative of the rotation number with respect to the action, vanishes. The twistless torus moves outward after it is created and eventually collides with the saddle-center bifurcation that creates the period three orbits. The existence of the twistless bifurcation is responsible for the breakdown of the nondegeneracy condition required in the proof of the KAM theorem for flows or the Moser twist theorem for maps. When the twistless torus has a rational rotation number, there are typically reconnection bifurcations of periodic orbits with that rotation number.

History

School

  • Science

Department

  • Mathematical Sciences

Pages

484391 bytes

Publication date

1999

Notes

This is a pre-print. The definitive version: DULLIN, H.R., MEISS, J.D., STERLING, D., 2000. Generic twistless bifurcations. Nonlinearity, 13 (1), pp.203-224, is available at: http://www.iop.org/EJ/journal/Non.

Language

en

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