04-410 Heinz Hanßmann

On Hamiltonian bifurcations of invariant tori with a Floquet multiplier -1 (871K, PostScript, gzipped and uuencoded) Dec 11, 04

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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.

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