Hamiltonian Hopf bifurcations in Gaudin models

TL;DR

This paper analyzes singularities in su(2) Gaudin models, revealing Hamiltonian Hopf bifurcations and classifying their nature using normal forms, with implications for the models' bifurcation structure.

Contribution

It provides a detailed normal form analysis of Hamiltonian Hopf bifurcations in Gaudin models, including degeneracy and bifurcation type classification.

Findings

Normal form for Hamiltonian Hopf bifurcation up to sixth order.

Criteria for degeneracy and supercritical/subcritical bifurcations.

Multiple bifurcation phenomena beyond Hamiltonian Hopf in Gaudin models.

Abstract

We show that su(2) rational and trigonometric Gaudin models, or in other words, generalised coupled angular momenta systems, have singularities that undergo Hamiltonian Hopf bifurcations. In particular, we find a normal form for the Hamiltonian Hopf bifurcation up to sixth order, letting us determine when the bifurcation is degenerate or not. Furthermore, in the non-degenerate case we may use the fourth order terms to determine whether the bifurcation is supercritical or subcritical; whether a flap appears in the image of the momentum map or not. Finally, figures illustrating some of the bifurcations taking place in su(2) Gaudin models are presented, showing that there are more bifurcations occurring than only Hamiltonian Hopf ones.