Generating hyperbolic singularities in completely integrable systems

TL;DR

This paper presents a method to locally modify integrable systems on symplectic 4-manifolds to generate hyperbolic singularities while preserving existing focus-focus singularities, expanding the types of singularities in such systems.

Contribution

It introduces a technique to create hyperbolic singularities in integrable systems without affecting existing focus-focus singularities, using Eliasson's linearization and Hamiltonian Hopf bifurcation.

Findings

Able to generate smooth curves of hyperbolic singularities

Preserves existing focus-focus singularities

Provides a local modification method for integrable systems

Abstract

Let $(M,\Omega)$ be a connected symplectic 4-manifold and let $F=(J,H) : M \to \mathbb{R}^2$ be a completely integrable system on $M$ with only non-degenerate singularities and for which $J : M \to \mathbb{R}$ is a proper map. Assume that $F$ does not have singularities with hyperbolic blocks and that $p_1,...,p_n$ are the focus-focus singularities of $F$. For each subset $S=\{i_1,...,i_j\}$ we will show how to modify $F$ locally around any $p_i, i \in S$, in order to create a new integrable system $\tilde{F}=(J, \tilde{H}) : M \to \mathbb{R}^2$ such that its classical spectrum $\tilde{F}(M)$ contains $j$ smooth curves of singular values corresponding to non-degenerate transversally hyperbolic singularities of $\tilde{F}$. Moreover the focus-focus singularities of $\tilde{F}$ are precisely $p_i$, $i \in \{1,...,n\} \setminus S$, and each of these $p_i$ is non-degenerate. The proof is based on Eliasson's linearization theorem for non-degenerate singularities, and properties of the Hamiltonian Hopf bifurcation.