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The \(1:\pm 2\) resonance. (English) Zbl 1229.37039

Summary: 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.

MSC:

37G10 Bifurcations of singular points in dynamical systems
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
37J20 Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
70K30 Nonlinear resonances for nonlinear problems in mechanics
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