15 research outputs found
Towards Hypersemitoric Systems
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.Comment: 26 pages, 8 figure
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
and coupled angular momenta on , 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 and proves that, for certain ranges of the parameters, it is a
compact semitoric system with precisely two focus-focus singularities. Since
the twisting index (one of the semitoric invariants) is related to the
relationship between different focus-focus points, this paper provides systems
for the future study of the twisting index.Comment: Update to most recent version: some typos removed; minor inaccuracies
corrected; better layou
Creating hyperbolic-regular singularities in the presence of an -symmetry
On a 4-dimensional compact symplectic manifold, we study how suitable
perturbations of a toric system to a family of completely integrable systems
with -symmetry lead to various hyperbolic-regular singularities.
We compute and visualise associated phenomena like flaps, swallowtails, and
-stacked tori for and give an upper bound for in our
family of systems.Comment: 44 pages, 22 figure
Survey on recent developments in semitoric systems
Semitoric systems are a special class of four-dimensional completely
integrable systems where one of the first integrals generates an
-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.Comment: 15 pages, 5 figure
Families of four-dimensional integrable systems with -symmetries
The aim of this paper is to give new insights about families of integrable
systems lifting a Hamiltonian -space. Specifically, we study one-parameter
families of systems with a fixed
Hamiltonian -space and which are semitoric for certain
values of the parameter , with a focus on such families in which one
singular point undergoes a Hamiltonian-Hopf bifurcation (also called nodal
trade in the context of semitoric systems, and more generally almost toric
fibrations). Beyond semitoric systems, we also study families containing
hypersemitoric systems, and we investigate the local theory of a nodal trade.
Building on and generalizing the ideas of a previous paper, we show how such
families can be used to find explicit semitoric systems with certain desired
invariants (bundled in the marked semitoric polygon). This allows us to make
progress on the semitoric minimal model program by understanding and coming up
with explicit systems for each strictly minimal type (i.e. those not admitting
any toric or semitoric type blowdown). In order to obtain these systems, we
develop strategies for constructing and understanding explicit examples of
semitoric (and hypersemitoric) systems in general. One strategy we make use of
is to start from a well-understood system (such as a toric system) and to
explicitly induce Hamiltonian-Hopf bifurcations to produce focus-focus singular
points. This is an expanded version of the technique used in the aforementioned
previous paper, in order to apply it to semitoric systems which include
non-trivial isotropy spheres in the underlying -space (i.e.
-spheres), which occurs in several of the strictly minimal
systems. In particular, we give an explicit one-parameter family of systems on
which transitions between being of toric type, semitoric type,
and hypersemitoric type depending on the value of the parameter. We study this
system at each stage, computing the marked semitoric polygon of the semitoric
system and determining several properties of the hypersemitoric system,
including the existence of a unique flap and two parabolic orbits. Furthermore,
we study the transitions between these stages. We also come up with new
explicit semitoric systems on all Hirzebruch surfaces which, together with the
previous systems and the systems already contained in the literature, gives an
explicit model for every type of strictly minimal system. Moreover, we show how
to obtain every strictly minimal system by applying sequences of alternating
toric type blowups and blowdowns to simple explicit systems. In particular, we
obtain that every strictly minimal semitoric system is part of a family
which is semitoric for all but a finite number of
values of , called a semitoric family.Comment: 145 pages, 53 figures. Comments welcome
Hamiltonian Monodromy and Morse Theory
We show that Hamiltonian monodromy of an integrable two degrees of freedom
system with a global circle action can be computed by applying Morse theory to
the Hamiltonian of the system. Our proof is based on Takens's index theorem,
which specifies how the energy-h Chern number changes when h passes a
non-degenerate critical value, and a choice of admissible cycles in
Fomenko-Zieschang theory. Connections of our result to some of the existing
approaches to monodromy are discussed
Open problems, questions, and challenges in finite-dimensional integrable systems
The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference “Finite-dimensional Integrable Systems, FDIS 2017” held at CRM, Barcelona in July 2017