 04410 Heinz Hanßmann
 On Hamiltonian bifurcations of invariant tori with a Floquet multiplier 1
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Dec 11, 04

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Abstract. Nearly integrable Hamiltonian systems are considered,
for which the unperturbed system has a lowerdimensional torus
not satisfying the second Mel'nikov condition ; on a 2:1 covering
space a suitable choice of the toral angles yields a vanishing
Floquet exponent. A nilpotent Floquet matrix $(^0_0 {}^1_0)$ leads
to the quasiperiodic analogue of the perioddoubling bifurcation,
so particular emphasis is given to the case $(^0_0 {}^0_0)$ of
vanishing normal linear behaviour. The actions conjugate to the
toral angles unfold the various ways in which the degenerate torus
becomes normally elliptic, hyperbolic or parabolic. With a
KAMtheoretic approach it is then shown that this bifurcation
scenario survives a nonintegrable perturbation, parametrised by
pertinent large Cantor sets.
The bifurcation scenario is governed by the `first' unimodal planar
singularity $\frac1{24}p^4 \pm \frac1{24} q^4 + \frac\mu{4} p^2 q^2$,
which has codimension 8 with respect to all planar singularities.
In the present context this high number is reduced to codimension 3
since the \pirotation on the 2:1 covering space has to be respected,
and in case the Hamiltonian system is reversible there is a further
reduction by 1 and the codimension becomes 2. In such low
codimensions it becomes more transparent why the modulus \mu 
although playing a prominent r\^ole during the KAM iteration  is
of limited influence on the dynamical implications.
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