Konstantinos Efstathiou

Metamorphoses of Hamiltonian Systems with Symmetries

Introduction.- Four Hamiltonian Systems.- Small Vibrations of Tetrahedral Molecules.- The Hydrogen Atom in Crossed Fields.- Quadratic Spherical Pendula.- Fractional Monodromy in the 1: - 2 Resonance System.- The Tetrahedral Group.- Local Properties of Equilibria.- References.- Index.

Efficient Structure-Aware Selection Techniques for 3D Point Cloud Visualizations with 2DOF Input

Data selection is a fundamental task in visualization because it serves as a pre-requisite to many follow-up interactions. Efficient spatial selection in 3D point cloud datasets consisting of thousands or millions of particles can be particularly challenging. We present two new techniques, TeddySelection and CloudLasso, that support the selection of subsets in large particle 3D datasets in an interactive and visually intuitive manner. Specifically, we describe how to spatially select a subset of a 3D particle cloud by simply encircling the target particles on screen using either the mouse or direct-touch input. Based on the drawn lasso, our techniques automatically determine a bounding selection surface around the encircled particles based on their density. This kind of selection technique can be applied to particle datasets in several application domains. TeddySelection and CloudLasso reduce, and in some cases even eliminate, the need for complex multi-step selection processes involving Boolean operations. This was confirmed in a formal, controlled user study in which we compared the more flexible CloudLasso technique to the standard cylinder-based selection technique. This study showed that the former is consistently more efficient than the latter - in several cases the CloudLasso selection time was half that of the corresponding cylinder-based selection.

Heteroclinic cycles between unstable attractors

We consider networks of pulse coupled linear oscillators with non-zero delay where the coupling between the oscillators is given by the Mirollo–Strogatz function. We prove the existence of heteroclinic cycles between unstable attractors for a network of four oscillators and for an open set of parameter values.

Escapes and recurrence in a simple Hamiltonian system

Many physical systems can be modeled as scattering problems. For example, the motions of stars escaping from a galaxy can be described using a potential with two or more escape routes. Each escape route is crossed by an unstable Lyapunov orbit. The region between the two Lyapunov orbits is where the particle interacts with the system. We study a simple dynamical system with escapes using a suitably selected surface of section. The surface of section is partitioned in different escape regions which are defined by the intersections of the asymptotic manifolds of the Lyapunov orbits with the surface of section. The asymptotic curves of the other unstable periodic orbits form spirals around various escape regions. These manifolds, together with the manifolds of the Lyapunov orbits, govern the transport between different parts of the phase space. We study in detail the form of the asymptotic manifolds of a central unstable periodic orbit, the form of the escape regions and the infinite spirals of the asymptotic manifolds around the escape regions. We compute the escape rate for different values of the energy. In particular, we give the percentage of orbits that escape after a finite number of iterations. In a system with escapes one cannot define a Poincaré recurrence time, because the available phase space is infinite. However, for certain domains inside the lobes of the asymptotic manifolds there is a finite ‘minimum recurrence time’. We find the minimum recurrence time as a function of the energy.

Classification of perturbations of the hydrogen atom by small static electric and magnetic fields

We consider perturbations of the hydrogen atom by sufficiently small homogeneous static electric and magnetic fields of all possible mutual orientations. Normalizing with regard to the Keplerian symmetry, we uncover resonances and conjecture that the parameter space of this family of dynamical systems is stratified into zones centred on the resonances. The 1 : 1 resonance corresponds to the orthogonal field limit, studied earlier by Cushman & Sadovskií (Cushman & Sadovskií 2000 Physica 142, 166–196). We describe the structure of the 1 : 1 zone, where the system may have monodromy of different kinds, and consider briefly the 1 : 2 zone.

Analysis of rotation-vibration relative equilibria on the example of a tetrahedral four atom molecule

We study relative equilibria (RE) of a nonrigid molecule, which vibrates about a well-defined equilibrium configuration and rotates as a whole. Our analysis unifies the theory of rotational and vib...

Robustness of unstable attractors in arbitrarily sized pulse-coupled networks with delay

We consider arbitrarily large networks of pulse-coupled oscillators with non-zero delay where the coupling is given by the Mirollo-Strogatz function. We prove that such systems have unstable attractors (saddle periodic orbits whose stable set has non-empty interior) in an open parameter region for three or more oscillators. The evolution operator of the system can be discontinuous and we propose an improved model with continuous evolution operator.

Perturbations of the 1:1:1 resonance with tetrahedral symmetry: a three degree of freedom analogue of the two degree of freedom Henon-Heiles Hamiltonian

We study a class of three degree of freedom (3-DOF) Hamiltonian systems that share certain characteristics with the 2-DOF Henon–Heiles Hamiltonian. Our systems represent a 1 : 1 : 1 resonant three-oscillator whose principal nonlinear perturbation is the cubic potential term xyz with tetrahedral symmetry. After normalizing and reducing the 1 : 1 : 1 oscillator symmetry, we show that near the limit of linearization all our systems can be described as a one-parametric family. Such reduced systems have been suggested earlier by Hecht (1960 J. Mol. Spectrosc. 5 355) and later by Patterson (1985 J. Chem. Phys. 83 4618) to model triply degenerate vibrations of tetrahedral molecules. We describe relative equilibria (RE) of these systems, classify all qualitatively different family members, and discuss bifurcations of RE involved in the transitions from one region of regular parameter values to the other.

Linear Hamiltonian Hopf bifurcation for point-group-invariant perturbations of the 1:1:1 resonance

We consider G×R–invariant Hamiltonians H on complex projective 2–space, where Gis a point group and R is the time–reversal group. We find the symmetry–induced stationary points of H and classify them in terms of their linear stability. We then determine those points that can undergo a linear Hamiltonian Hopf bifurcation.

CAST: Effective and Efficient User Interaction for Context-Aware Selection in 3D Particle Clouds

We present a family of three interactive Context-Aware Selection Techniques (CAST) for the analysis of large 3D particle datasets. For these datasets, spatial selection is an essential prerequisite to many other analysis tasks. Traditionally, such interactive target selection has been particularly challenging when the data subsets of interest were implicitly defined in the form of complicated structures of thousands of particles. Our new techniques SpaceCast, TraceCast, and PointCast improve usability and speed of spatial selection in point clouds through novel context-aware algorithms. They are able to infer a users subtle selection intention from gestural input, can deal with complex situations such as partially occluded point clusters or multiple cluster layers, and can all be fine-tuned after the selection interaction has been completed. Together, they provide an effective and efficient tool set for the fast exploratory analysis of large datasets. In addition to presenting Cast, we report on a formal user study that compares our new techniques not only to each other but also to existing state-of-the-art selection methods. Our results show that Cast family members are virtually always faster than existing methods without tradeoffs in accuracy. In addition, qualitative feedback shows that PointCast and TraceCast were strongly favored by our participants for intuitiveness and efficiency.