Bifurcations and monodromy of the axially symmetric 1:1:−2 resonance

Konstantinos Efstathiou; Heinz Hanßmann; Antonella Marchesiello

doi:10.1016/j.geomphys.2019.103493

Bifurcations and monodromy of the axially symmetric 1:1:−2 resonance

Konstantinos Efstathiou^*, Heinz Hanßmann, Antonella Marchesiello

^*Corresponding author for this work

University of Groningen

Research output: Contribution to journal › Article › Academic › peer-review

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.

Original language	English
Article number	103493
Journal	Journal of Geometry and Physics
Volume	146
DOIs	https://doi.org/10.1016/j.geomphys.2019.103493
Publication status	Published - 1 Dec 2019

Funding

A.M. was supported by INdAM (Istituto Nazionale di Alta Matematica “F. Severi”) and by the Grant Agency of the Czech Republic, project 17-11805S. Appendix

Keywords

Bifurcations
Hamiltonian monodromy
Reduction
Resonance

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10.1016/j.geomphys.2019.103493

1-s2.0-S0393044019301743-mainFinal published version, 2.98 MBLicence: Taverne

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AB - 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.

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KW - Resonance

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Bifurcations and monodromy of the axially symmetric 1:1:−2 resonance

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Keywords

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