arXiv subjects

Otto van Koert

Publications and source records attributed to Otto van Koert.

At least 19 recordsLinked to original sources

arXiv2026

Bifurcations of highly inclined near halo orbits using Moser regularization

We study the bifurcation structure of highly inclined near halo orbits with close approaches to the light primary, in the circular restricted three-body problem (CR3BP). Using a Hamiltonian formulation together with Moser regularization, we develop a numerical framework for the continuation of periodic orbits and the computation of their Floquet multipliers which remains effective near collision. We describe vertical collision orbits and families emerging from its pitchfork, period-doubling, and period-tripling bifurcations in the limiting Hill's problem, including the halo and butterfly families. We continue these into the CR3BP using a perturbative framework via a symplectic scaling, and construct bifurcation graphs for representative systems (Saturn-Enceladus, Earth-Moon, Copenhagen) to identify common dynamical features. Conley-Zehnder indices are computed to classify the resulting families. Together, these results provide a coherent global picture of polar orbit architecture near the light primary, offering groundwork for future mission design, such as Enceladus plume sampling missions.

math.DS↗

arXiv2025

Computational symplectic topology and symmetric orbits in the restricted three-body problem

In this paper we propose a computational approach to proving the Birkhoff conjecture on the restricted three-body problem, which asserts the existence of a disk-like global surface of section. Birkhoff had conjectured this surface of section as a tool to prove existence of a direct periodic orbit. Using techniques from validated numerics we prove the existence of an approximately circular direct orbit for a wide range of mass parameters and Jacobi energies. We also provide methods to rigorously compute the Conley-Zehnder index of periodic Hamiltonian orbits using computational tools, thus giving some initial steps for developing computational Floer homology and providing the means to prove the Birkhoff conjecture via symplectic topology. We apply this method to various symmetric orbits in the restricted three-body problem.

math.SG↗

arXiv2024

Deep learning based ECG segmentation for delineation of diverse arrhythmias

Accurate delineation of key waveforms in an ECG is a critical step in extracting relevant features to support the diagnosis and treatment of heart conditions. Although deep learning based methods using segmentation models to locate P, QRS, and T waves have shown promising results, their ability to handle arrhythmias has not been studied in any detail. In this paper we investigate the effect of arrhythmias on delineation quality and develop strategies to improve performance in such cases. We introduce a U-Net-like segmentation model for ECG delineation with a particular focus on diverse arrhythmias. This is followed by a post-processing algorithm which removes noise and automatically determines the boundaries of P, QRS, and T waves. Our model has been trained on a diverse dataset and evaluated against the LUDB and QTDB datasets to show strong performance, with F1-scores exceeding 99% for QRS and T waves, and over 97% for P waves in the LUDB dataset. Furthermore, we assess various models across a wide array of arrhythmias and observe that models with a strong performance on standard benchmarks may still perform poorly on arrhythmias that are underrepresented in these benchmarks, such as tachycardias. We propose solutions to address this discrepancy.

cs.LG↗

arXiv2024

Symplectic geometry and space mission design

Using methods from symplectic geometry, the second and fifth authors have provided theoretical groundwork and tools aimed at analyzing periodic orbits, their stability and their bifurcations in families, for the purpose of space mission design. The Broucke stability diagram was refined, and the "Floer numerical invariants" where considered, as numbers which stay invariant before and after a bifurcation, and therefore serve as tests for the algorithms used. These tools were later employed for numerical studies. In this article, we will further illustrate these methods with numerical studies of families of orbits for the Jupiter-Europa and Saturn-Enceladus systems, with emphasis on planar-to-spatial bifurcations, from deformation of the families in Hill's lunar problem studied by the first author. We will also provide an algorithm for the numerical computation of Conley--Zehnder indices, which are instrumental in practice for determining which families of orbits connect to which. As an application, we use our tools to study a family of periodic orbits that approaches Enceladus at an altitude of 29km, and therefore may be used in future space missions to visit the water plumes.

math.SG↗

arXiv2022

Expectation of topological invariants

In this paper, we study the expectation values of topological invariants of the Vietoris-Rips complex and Čech complex for a finite set of sample points on a Riemannian manifold. We show that the Betti number and Euler characteristic of the complexes are Lipschitz functions of the scale parameter and that there is an interval such that the Betti curve converges to the Betti number of the underlying manifold.

math.GT↗

arXiv2022

Reeb flows without simple global surfaces of section

We construct, for any given positive integer $n$, Reeb flows on contact integral homology 3-spheres which do not admit global surfaces of section with fewer than $n$ boundary components. We use a connected sum operation for open books to construct such systems. We prove that this property is stable with respect to $C^{4+ε}$-small perturbations of the Hamiltonian given on the symplectization.

math.SG↗

arXiv2022

Global hypersurfaces of section in the spatial restricted three-body problem

We propose a contact-topological approach to the spatial circular restricted three-body problem, for energies below and slightly above the first critical energy value. We prove the existence of a circle family of global hypersurfaces of section for the regularized dynamics. Below the first critical value, these hypersurfaces are diffeomorphic to the unit disk cotangent bundle of the $2$-sphere, and they carry symplectic forms on their interior, which are each deformation equivalent to the standard symplectic form. The boundary of the global hypersurface of section is an invariant set for the regularized dynamics that is equal to a level set of the Hamiltonian describing the regularized planar problem. The first return map is Hamiltonian, and restricts to the boundary as the time-$1$ map of a positive reparametrization of the Reeb flow in the planar problem. This construction holds for any choice of mass ratio, and is therefore non-perturbative. We illustrate the technique in the completely integrable case of the rotating Kepler problem, where the return map can be studied explicitly.

math.SG↗

arXiv2021

A generalized Poincaré-Birkhoff theorem

We prove a generalization of the classical Poincaré--Birkhoff theorem for Liouville domains, in arbitrary even dimensions. This is inspired by the existence of global hypersurfaces of section for the spatial case of the restricted three-body problem (as proved by the authors in arXiv:2011.10386).

math.SG↗

arXiv2019

A Reeb flow on the three-sphere without a disk-like global surface of section

We show that there are Reeb flows on the standard, tight three-sphere that do not admit global surfaces of section with one boundary component. In particular, the Reeb flows that we construct do not admit disk-like global surfaces of section. These Reeb flows are constructed using integrable systems, and a connected sum construction that extends the integrable system.

math.SG↗

arXiv2018

The Omega limit set of a family of chords

In this paper we study the limit behavior of a family of chords on compact energy hypersurfaces of a family of Hamiltonians. Under the assumption that the energy hypersurfaces are all of contact type, we give results on the Omega limit set of this family of chords. Roughly speaking, such a family must either end in a degeneracy, in which case it joins another family, or can be continued. This gives a Floer theoretic explanation of the behavior of certain families of symmetric periodic orbits in many well-known problems, including the restricted three-body problem.

math.SG↗

arXiv2018

A Family of Periodic Orbits in the Three-Dimensional Lunar Problem

A family of periodic orbits is proven to exist in the spatial lunar problem that are continuations of a family of consecutive collision orbits, perpendicular to the primary orbit plane. This family emanates from all but two energy values. The orbits are numerically explored. The global properties and geometry of the family is studied.

math.DS↗

arXiv2017

Periodic orbits in the restricted three-body problem and Arnold's $J^+$-invariant

We apply Arnold's theory of generic smooth plane curves to Stark-Zeeman systems. This is a class of Hamiltonian dynamical systems that describes the dynamics of an electron in an external electric and magnetic field, and includes many systems from celestial mechanics. Based on Arnold's $J^+$-invariant, we introduce invariants of periodic orbits in planar Stark-Zeeman systems and study their behaviour.

math.SG↗

arXiv2016

A convex embedding for the rotating Kepler problem

In this note, we prove that below the first critical energy level, a proper combination of the Ligon-Schaaf and Levi-Civita regularization mappings provides a convex symplectic embedding of the energy surfaces of the planar rotating Kepler problem into $R^4$ endowed with its standard symplectic structure. A direct consequence is the dynamical convexity of the planar rotating Kepler problem, which has been established by Albers-Fish-Frauenfelder-van Koert by direct computations. This result opens up new approaches to attack the Birkhoff conjecture about the existence of a global surface of section in the restricted planar circular three body problem using holomorphic curve techniques.

math.SG↗

arXiv2015

Brieskorn Manifolds, Positive Sasakian Geometry, and Contact Topology

Using $S^1$-equivariant symplectic homology, in particular its mean Euler characteristic, of the natural filling of links of Brieskorn-Pham polynomials, we prove the existence of infinitely many inequivalent contact structures on various manifolds, including in dimension 5 the k-fold connected sums of $S^2\times S^3$ and certain rational homology spheres. We then apply our result to show that on these manifolds the moduli space of classes of positive Sasakian structures has infinitely many components. We also apply our results to give lower bounds on the number of components of the moduli space of Sasaki-Einstein metrics on certain homotopy spheres. Finally a new family of Sasaki-Einstein metrics of real dimension 20 on $S^5$ is exhibited.

math.DG↗

arXiv2015

Brieskorn manifolds in contact topology

In this survey, we give an overview of Brieskorn manifolds and varieties, and their role in contact topology. We discuss open books, fillings and invariants such as contact and symplectic homology. We also present some new results involving exotic contact structures, invariants and orderability. The main tool for the required computations is a version of the Morse-Bott spectral sequence. We provide a proof for the particular version that is useful for us.

math.SG↗

arXiv2015

Contact open books with exotic pages

We consider a fixed contact 3-manifold that admits infinitely many compact Stein fillings which are all homeomorphic but pairwise non-diffeomorphic. Each of these fillings gives rise to a closed contact 5-manifold described as a contact open book whose page is the filling at hand and whose monodromy is the identity symplectomorphism. We show that the resulting infinitely many contact 5-manifolds are all diffeomorphic but pairwise non-contactomorphic. Moreover, we explicitly determine these contact 5-manifolds.

math.GT↗

arXiv2014

The torsion flow on a closed pseudohermitian 3-manifold

In this paper we define the torsion flow, a CR analogue of the Ricci flow. For homogeneous CR manifolds we give explicit solutions to the torsion flow illustrating various kinds of behavior. We also derive monotonicity formulas for CR entropy functionals. As an application, we classify torsion breathers.

math.DG↗

arXiv2014

Symmetric periodic orbits and uniruled real Liouville domains

A real Liouville domain is a Liouville domain together with an exact anti-symplectic involution. We call a real Liouville domain uniruled if there exists an invariant finite energy plane through every real point. Asymptotically an invariant finite energy plane converges to a symmetric periodic orbit. In this note we work out a criterion which guarantees uniruledness for real Liouville domains.

math.SG↗