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(Vanishing) Twist in the Saddle-Centre and Period-Doubling Bifurcation
The lowest order resonant bifurcations of a periodic orbit of a Hamiltonian
system with two degrees of freedom have frequency ratio 1:1 (saddle-centre) and
1:2 (period-doubling). The twist, which is the derivative of the rotation
number with respect to the action, is studied near these bifurcations. When the
twist vanishes the nondegeneracy condition of the (isoenergetic) KAM theorem is
not satisfied, with interesting consequences for the dynamics. We show that
near the saddle-centre bifurcation the twist always vanishes. At this
bifurcation a ``twistless'' torus is created, when the resonance is passed. The
twistless torus replaces the colliding periodic orbits in phase space. We
explicitly derive the position of the twistless torus depending on the
resonance parameter, and show that the shape of this curve is universal. For
the period doubling bifurcation the situation is different. Here we show that
the twist does not vanish in a neighborhood of the bifurcation.Comment: 18 pages, 9 figure
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