Heinz Hanßmann
On Hamiltonian bifurcations of invariant tori with a Floquet multiplier -1
(871K, PostScript, gzipped and uuencoded)
ABSTRACT. Nearly integrable Hamiltonian systems are considered,
for which the unperturbed system has a lower-dimensional 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 quasi-periodic analogue of the period-doubling 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
KAM-theoretic approach it is then shown that this bifurcation
scenario survives a non-integrable 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 co-dimension 8 with respect to all planar singularities.
In the present context this high number is reduced to co-dimension 3
since the \pi-rotation 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 co-dimension becomes 2. In such low
co-dimensions 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.