A note on focus-focus singularities

Parabolicity of degenerate singularities of axisymmetric Zhukovsky case

The degenerate singularities of systems from one well-known multiparameter family of integrable systems of rigid body dynamics are studied. Axisymmetric Zhukovsky systems are considered, i.e. axisymmetric Euler tops after adding a constant gyrostatic moment. For all values of the set of parameters, excluding some hypersurfaces, it is proved that the degenerate local and semi-local singularities of the system are of the parabolic and cuspidal type, respectively. Thus these singularities are structurally stable under small perturbations of the system in the class of integrable systems. Note that…

含两个焦点-焦点奇点的四参数半环面系统的高度不变量

The height invariant of a four-parameter semitoric system with two focus-focus singularities

本文针对含两个焦点-焦点奇点的四参数半环面系统，计算了其高度不变量（除多边形不变量外），并证明该不变量的两个分量以复杂方式编码了系统的对称性，推进了多参数半环面系统不变量的计算研究。

A family of compact semitoric systems with two focus-focus singularities

About 6 years ago, semitoric systems were classified by Pelayo & Vu Ngoc by means of five invariants. Standard examples are the coupled spin oscillator on $\mathbb{S}^2 \times \mathbb{R}^2$ and coupled angular momenta on $\mathbb{S}^2 \times \mathbb{S}^2$, both having exactly one focus-focus singularity. But so far there were no explicit examples of systems with more than one focus-focus singularity which are semitoric in the sense of that classification. This paper introduces a 6-parameter family of integrable systems on $\mathbb{S}^2 \times \mathbb{S}^2$ and proves that, for certain ranges o…

A family of semitoric systems with four focus-focus singularities and two double pinched tori

We construct a 1-parameter family $F_t=(J, H_t)_{0 \leq t \leq 1}$ of integrable systems on a compact $4$-dimensional symplectic manifold $(M, ω)$ that changes smoothly from a toric system $F_0$ with eight elliptic-elliptic singular points via toric type systems to a semitoric system $F_t$ for $ t^- < t < t^+$. These semitoric systems $F_t$ have precisely four elliptic-elliptic and four focus-focus singular points. Moreover, at $t= \frac{1}{2}$, the system has precisely two focus-focus fibres each of which contains exactly two focus-focus points, giving these fibres the shape of double pinched…

Generating hyperbolic singularities in completely integrable systems

Let $(M,Ω)$ 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 $\tilde{F}(M)$ conta…