Bifurcations and Monodromy of the Axially Symmetric 1:1:-2 Resonance

K. Efstathiou, H. Hanßmann, and A. Marchesiello
Arxiv preprint

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Abstract

We consider integrable Hamiltonian systems in three degrees of freedom near an elliptic equilibrium in 1:1:-2 resonance. The integrability originates from averaging along the periodic motion of the quadratic part and an imposed rotational symmetry about the vertical axis. Introducing a detuning parameter we find a rich bifurcation diagram, containing three parabolas of Hamiltonian Hopf bifurcations that join at the origin. We describe the monodromy of the resulting ramified 3-torus bundle as variation of the detuning parameter lets the system pass through 1:1:-2 resonance.