Bifurcations and monodromy of the axially symmetric \(1:1:-2\) resonance. (English) Zbl 1469.37042
Summary: 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.
MSC:
| 37J20 | Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems |
| 37J39 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.) |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
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