arXiv subjects
Publications and source records attributed to Sonja Hohloch.
We adapt and generalize McDuff's method of probes from toric system to so called semitoric integrable systems and apply it to study the (non)displaceability properties of the fibers of $3$ examples of semitoric integrable systems.
Given any compact connected four dimensional symplectic manifold $(M,ω)$ and smooth function $J\colon M\to \mathbb{R}$ which generates an effective $\mathbb{S}^1$-action, we show that there exists a smooth function $H\colon M\to\mathbb{R}$ such that $(M,ω,(J,H))$ is a completely (Liouville) integrable system of a type we call hypersemitoric -- these are systems for which all singularities are non-degenerate, except possibly for a finite number of families of degenerate points of a relatively tame type called parabolic (also sometimes called cuspidal). Such an $(M,ω,J)$ is often referred to as a Hamiltonian $\mathbb{S}^1$-space (classified by Karshon in 1999) and we call any integrable system of the form $(M,ω,(J,H))$ an extension of $(M,ω,J)$. Using this terminology, our main result is that any Hamiltonian $\mathbb{S}^1$-space can be extended to a hypersemitoric integrable system. We also show that there exist Hamiltonian $\mathbb{S}^1$-spaces for which any extension must include at least one degenerate singular point. Parabolic points are among the most common and natural degenerate points, and thus hypersemitoric systems are in this sense the `nicest' class of systems to which all Hamiltonian $\mathbb{S}^1$-spaces can be extended. We also prove several foundational results about these systems, such as the non-existence of loops of hyperbolic-regular points and some properties about their fibers.
The notion of an exterior differential system (on a manifold) has recently been extended to the setting of a Lie algebroid. Here, we further develop the theory and we present two versions of the Cartan-Kähler theorem in the case where the anchor map of the Lie algebroid is surjective. We give an illustrative example and, as a concrete application, we make use of our results in a specific case of the so-called invariant inverse problem of the calculus of variations.
Hypersemitoric systems are a class of integrable systems on $4$-dimensional symplectic manifolds which only have mildly degenerate singularities and where one of the integrals induces an effective Hamiltonian $S^1$-action and is proper. We introduce the affine invariant of hypersemitoric systems, which is a generalization of the Delzant polytope of toric systems and the polytope invariant of semitoric systems. Along the way, we compute and plot this invariant for meaningful and more and more complicated examples.
In this paper we introduce a new bifurcation in Hamiltonian systems, which we call the double flip bifurcation. The Hamiltonian depends on two parameters, one of which controls the double flip bifurcation. The result of the bifurcation is the occurrence of two Hamiltonian flip bifurcations with respect to the other parameter. The two Hamiltonian flip bifurcations are simultaneous with respect to the first parameter, and are connected by a curve-segment of singular points. We find a normal form for Hamiltonians describing systems going through double flip bifurcations, and compute said normal form for some examples.
Assume $M$ to be $\mathbb R^2$ or a closed surface of genus $g \geq 1$ and $ω$ a symplectic form on $M$. Let $φ: M \to M$ be a symplectomorphism with hyperbolic fixed point $x$ and transversely intersecting stable and unstable manifolds $W^s(φ, x)$ and $ W^u(φ, x)$. The intersection points $W^s(φ, x) \cap\ W^u(φ, x)=:{\mathcal H}(φ, x)$ are called homoclinic points, and the (un)stable manifolds of a symplectomorphism are Lagrangian submanifolds. For this Lagrangian intersection problem with its wildly oscillating Lagrangian manifolds and infinite number of intersection points, we introduced in earlier works Floer homologies generated by so-called (semi)primary homoclinic points and analysed their dynamical and geometric properties. In this paper, we significantly generalise these earlier results by first defining a Floer homology generated by finite sets of contractible homoclinic points. These Floer homologies nevertheless still consider rather `local' aspects of $W^s(φ, x) \cap\ W^u(φ, x)$ since their generator sets are finite (but the number of contractible homoclinic points is infinite). To overcome this issue, we construct a direct limit of these `local' homoclinic Floer homologies over suitable index sets. These direct limits accumulate the information gathered by the finitely generated `local' homoclinic Floer homologies.
These lecture notes grew out of notes for courses around Integrable PDEs and the KdV equation given by the authors during the past five years at the University of Antwerp (Belgium). Comments and suggestions are welcome.
Semitoric integrable systems were symplectically classified by Pelayo and Vu Ngoc in 2009-2011 in terms of five invariants. Four of these invariants were already well-understood prior to the classification, but the fifth invariant, the so-called twisting index invariant, came as a surprise. Intuitively, the twisting index encodes how the structure in a neighborhood of a focus-focus fiber compares to the large-scale structure of the semitoric system and it was originally defined by comparing certain momentum maps. In the first half of the present paper, we produce several new formulations of the twisting index which give rise to dynamical, geometric, and topological interpretations. More specifically, we describe it in terms of differences of action variables, Taylor series, and homology cycles. In the second half of the paper, we compute the twisting index invariant of a specific family of systems with two focus-focus singular points (the so-called generalized coupled angular momenta), which is the first time that the twisting index has been computed for a system with more than one focus-focus point. Moreover, we also compute the terms of the Taylor series invariant up to second order. Since the other invariants of this family were already computed, this becomes the third family of semitoric systems for which all invariants are known, after the coupled spin oscillators and the coupled angular momenta.
Hypersemitoric systems are 2-degree-of-freedom integrable systems on 4-dimensional manifolds that have an underlying $S^1$-symmetry and no degenerate singularities apart from maybe a finite number of families of so-called parabolic singularities. We give a short overview of recent examples displaying various bifurcations and sketch a topological-combinatorial classification of the connected components of fibers of hypersemitoric systems.
In this paper, we provide tools to study the dynamics of point vortex dynamics on $\mathbb{CP}^n$ and the flag manifold $\mathbb{F}_{1,2}(\mathbb{C}^3)$. These are the only Kähler twistor spaces arising from 4-manifolds. We give an explicit expression for Green's function on $\mathbb{CP}^n$ which enables us to determine the Hamiltonian $H$ and the equations of motions for the point vortex problem on $\mathbb{CP}^n$. Moreover, we determine the momentum map $μ:\mathbb{F}_{1,2}(\mathbb{C}^3)\to \mathfrak{su}^*(3)$ on the flag manifold.
This survey gives a short and comprehensive introduction to a class of finite-dimensional integrable systems known as hypersemitoric systems, recently introduced by Hohloch and Palmer in connection with the solution of the problem how to extend Hamiltonian circle actions on symplectic 4-manifolds to integrable systems with `nice' singularities. The quadratic spherical pendulum, the Euler and Lagrange tops (for generic values of the Casimirs), coupled-angular momenta, and the coupled spin oscillator system are all examples of hypersemitoric systems. Hypersemitoric systems are a natural generalization of so-called semitoric systems (introduced by Vu Ngoc) which in turn generalize toric systems. Speaking in terms of bifurcations, semitoric systems are `toric systems with/after supercritical Hamiltonian-Hopf bifurcations'. Hypersemitoric systems are `semitoric systems with, among others, subcritical Hamiltonian-Hopf bifurcations'. Whereas the symplectic geometry and spectral theory of toric and semitoric sytems is by now very well developed, the theory of hypersemitoric systems is still forming its shape. This short survey introduces the reader to this developing theory by presenting the necessary notions and results as well as its connections to other areas of mathematics and mathematical physics.
In this article, we introduce $b$-semitoric systems as a generalization of semitoric systems, specifically tailored for $b$-symplectic manifolds. The objective of this article is to furnish a collection of examples and investigate the distinctive characteristics of these systems. A $b$-semitoric system is a 4-dimensional $b$-integrable system that satisfies certain conditions: one of its momentum map components is proper and generates an effective global $S^1$-action, and all singular points are non-degenerate and devoid of hyperbolic components. To illustrate this concept, we provide five examples of $b$-semitoric systems by modifying the coupled spin oscillator and the coupled angular momenta, and we also classify their singular points. Additionally, we describe the dynamics of these systems through the image of their respective momentum maps.
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.
In this paper we present a symplectic analogue of the Fueter theorem. This allows the construction of special (polynomial) solutions for the symplectic Dirac operator $D_s$, which is defined as the first-order $\mathfrak{sp}(2n)$-invariant differential operator acting on functions on ${\mathbb R}^{2n}$ taking values in the metaplectic spinor representation.
Semitoric systems are a special class of completely integrable systems with two degrees of freedom that have been symplectically classified by Pelayo and Vu Ngoc about a decade ago in terms of five symplectic invariants. If a semitoric system has several focus-focus singularities, then some of these invariants have multiple components, one for each focus-focus singularity. Their computation is not at all evident, especially in multi-parameter families. In this paper, we consider a four-parameter family of semitoric systems with two focus-focus singularities. In particular, apart from the polygon invariant, we compute the so-called height invariant. Moreover, we show that the two components of this invariant encode the symmetries of the system in an intricate way.
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 tori. We exemplarily parametrise one of these fibres explicitly.
Semitoric systems are a special class of four-dimensional completely integrable systems where one of the first integrals generates an $\mathbb{S}^1$-action. They were classified by Pelayo & Vu Ngoc in terms of five symplectic invariants about a decade ago. We give a survey over the recent progress which has been mostly focused on the explicit computation of the symplectic invariants for families of semitoric systems depending on several parameters and the generation of new examples with certain properties, such as a specific number of singularities of lowest rank.
We characterize completely integrable Hamiltonian systems inducing an effective Hamiltonian torus action as systems with zero transport costs w.r.t. the time-$T$ map where $T \in {\mathbb R}^n$ is the period of the acting $n$-torus.