This article is cited in 16 papers

On the 65th birthday of R.Cushman

The $1 : \pm 2$ Resonance

R.H. Cushman^ab, H. R. Dullin^c, H.Hanßmann^a, S. Schmid<sup>c

a</sup> Mathematics Institute, University of Utrecht, P.O.Box 80.010, 3508 TA Utrecht, The Netherlands
^b Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, T2N1N4 Canada
^c Department of Mathematical Sciences, Loughborough University, LE11 3TU, UK

Abstract: On the linear level elliptic equilibria of Hamiltonian systems are mere superpositions of harmonic oscillators. Non-linear terms can produce instability, if the ratio of frequencies is rational and the Hamiltonian is indefinite. In this paper we study the frequency ratio $\pm 1/2$ and its unfolding. In particular we show that for the indefinite case $(1:-2)$ the frequency ratio map in a neighborhood of the origin has a critical point, i.e. the twist condition is violated for one torus on every energy surface near the energy of the equilibrium. In contrast, we show that the frequency map itself is non-degenerate (i.e. the Kolmogorov non-degeneracy condition holds) for every torus in a neighborhood of the equilibrium point. As a by product of our analysis of the frequency map we obtain another proof of fractional monodromy in the $1:-2$ resonance.

Keywords: resonant oscillators, normal form, singular reduction, bifurcation, energy-momentum mapping, monodromy.

MSC: 37G10, 37J15, 37J20, 37J35, 70K30

Received: 14.08.2007
Accepted: 12.10.2007

Language: English

DOI: 10.1134/S156035470706007X

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