Creating hyperbolic-regular singularities in the presence of an $\mathbb{S}^1$-symmetry

TL;DR

This paper investigates how perturbations of toric systems on 4D symplectic manifolds with $ ext{S}^1$-symmetry generate hyperbolic-regular singularities, visualizing complex phenomena like swallowtails and stacked tori.

Contribution

It introduces a method to produce and analyze hyperbolic-regular singularities in integrable systems with symmetry, including visualizations and bounds for singularity types.

Findings

Identification of hyperbolic-regular singularities through perturbations

Visualization of phenomena like flaps, swallowtails, and stacked tori

Establishment of an upper bound for the complexity of singularities

Abstract

On a 4-dimensional compact symplectic manifold, we study how suitable perturbations of a toric system to a family of completely integrable systems with $\mathbb{S}^1$-symmetry lead to various hyperbolic-regular singularities. We compute and visualise associated phenomena like flaps, swallowtails, and $k$-stacked tori for $k \in \{2, 3, 4\}$ and give an upper bound for $k$ in our family of systems.