## Generic twistless bifurcations

preprint

posted on 16.02.2006, 14:34 by Holger R. Dullin, J.D. Meiss, D.G. SterlingWe 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.

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