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

TL;DR

This paper analyzes the complex bifurcation structure and monodromy phenomena in a three-degree-of-freedom integrable Hamiltonian system near a 1:1:-2 resonance, revealing rich bifurcation patterns and topological changes.

Contribution

It introduces a detailed bifurcation diagram for the 1:1:-2 resonance and describes the monodromy of the associated 3-torus bundle as parameters vary.

Findings

Identification of three Hamiltonian Hopf bifurcation parabolas

Description of monodromy in the ramified 3-torus bundle

Analysis of bifurcation diagram near the 1:1:-2 resonance

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.