Potential in the “Clearance” of torus: Development on spherical functions

The method of solution of the Dirichlet problem for potentials of volume bodies with torus topology when the boundary conditions are set in the form of a series on spherical harmonics on pieces of two spherical surfaces is developed. The problem representation of an exterior potential homogeneous gravitating (or charged by static electric charge) circular torus out of a substance in a special (??intermediate??) spherical zone is presented and solved. The solution is received in the form of a combination of the Laplace series on even positive and odd negative degrees of the radius-vector of a test point. Coefficients of this series are received in a final analytical form. The general member of the series at the limit of big n tends to zero so the series converges fast and the radius of convergence are defined by torus geometry. The specified solution meets a gap in the theory, connecting together earlier discovered by us, two expansions in the Laplace series of a torus potential in ??interior?? and ??exterior?? spherical space zones. Thus, it is proved that it is possible to present the torus potential by power rows in all free space from the substance. For control of results by means of the obtained series, equipotential surfaces of the torus were calculated.     

Gravitational and electrostatic fields of a homogeneous circular cone

The problem of the simplest elements forming the external field of a gravitating (or electrostatically charged) homogeneous circular cone is formulated and solved by the method of contour integrals. It is found that the tip point has a logarithmic singularity and is not included in the equigravitating frame of the figure. On the whole, only one equigravitating rod with a purely imaginary density described by elementary functions exists for a cone. It is proved that this rod satisfies all necessary requirements: its mass and spatial potential are real-valued and equivalent to analogous characteristics of the cone. Independent verification in the asymptotic limit of an inhomogeneous circular disk confirms the main result. The spatial potential of the cone is expressed, using the rod, first in terms of a single integral and then in terms of standard full elliptic integrals, as well as a special series in harmonic functions. A family of equipotential surfaces is obtained.     

Polynomial integrals of mechanical systems on a torus with a singular potential

The problem on integrability of the equations of motion of a material point on an n-dimensional Euclidean torus under the action of a force field with the potential energy having singularities at a finite number of points is considered. It is assumed that these singularities contain logarithmic coefficients and, consequently, have a more general form in comparison with power features. The potentials having power-type singularities were considered previously by V.V. Kozlov and D.V. Treshchev. In this work, it is proved that the equations of motion in the problem under consideration admit no nontrivial momentum-polynomial first integral with integrable coefficients on this torus.     

Homoclinic tangency and chaotic attractor disappearance in a dripping faucet experiment

A sequence of attractors, reconstructed from interdrops time series data of a leaky faucet experiment, is analyzed as a function of the mean dripping rate. We established the presence of a saddle point and its manifolds in the attractors and we explained the dynamical changes in the system using the evolution of the manifolds of the saddle point, as suggested by the orbits traced in first return maps. The sequence starts at a fixed point and evolves to an invariant torus of increasing diameter (a Hopf bifurcation) that pushes the unstable manifold towards the stable one. The torus breaks up and the system shows a chaotic attractor bounded by the unstable manifold of the saddle. With the attractor expansion the unstable manifold becomes tangential to the stable one, giving rise to the sudden disappearance of the chaotic attractor, which is an experimental observation of a so called chaotic blue sky catastrophe.     

Tailoring of the flip effect in the orientation of a magnet levitating over a superconducting torus: Geometrical dependencies

In a previous study, a general local model was used in order to demonstrate the apparition of a flip effect in the equilibrium orientation of a magnet when it is over a superconducting torus. This effect can be easily used in devices such as binary position detectors for magneto-microscopy, contactless sieves or magnetic levels amongst others.We present an initial study useful to design devices based on the flip effect between magnets and torus superconductors. It demonstrates that varying different geometrical parameters the flip effect point can be fixed. Also, it can be observed that increasing the inner radius of the torus elevates the flip effect point. A magneto-mechanical explanation of this phenomenon is exposed. For an increment of cross-section diameter occurs the same behavior. There are linear piecewises in the geometrical dependency functions that can be used for a more accurate fitting of the flip effect point.     

Laplace series expansion of the internal potential of a homogeneous circular torus

The internal potential of a homogeneous circular torus first is represented by a series expansion in spherical functions (Laplace series). Exact analytical formulas for the coefficients of this series are derived and it is shown that they can be expressed through the standard Gauss hypergeometric function depending only on the geometric parameter of the torus. Convergence of the series is proved and the radius of convergence is determined. The relation of the radius with the torus geometrical parameter is found. A special spherical shell, where the problem of expansion of the torus potential should be solved in additional investigations, is detected.     

Geometry of two dimensional tori in phase space: Projections,sections and the Wigner function

The invariant manifolds (or “classical eigenstates”) in the phase space of bound integrable dynamical systems are known to be tori. Sections and projections of general, and special, two dimensional tori in four dimensional phase space are considered. Particular attention is paid to the families of projections accessed by linear canonical transformation since these can (in a certain sense) be considered to be different views of the same torus. The Wigner phase space representation of the corresponding semiclassical quantum eigenstate for a torus of any dimensionality is examined following the analysis of M. V. Berry (Phil. Trans. Roy. Soc.287 (1977), 237) for one dimensional tori. In this, the value of the semiclassical Wigner function at any phase space point depends on the behaviour of the chords of the torus centred on that point. It is found that for a two dimensional torus the number of such chords is always even. The three dimensional surfaces across which the number of chords changes constitute a (double) fold catastrophe on which the function oscillates with large amplitude. On the torus manifold itself this “Wigner caustic” generally exhibits a hyperbolic umbilic singularity (possibly interspersed with elliptic regions). At special lines and points on the torus, however, higher catastrophes up to E₈ are generic.     

Non-ergodicity of Two Particles Interacting via a Smooth Potential

We examine two point particles interacting via a smooth Lennard-Jones-type potential of finite range on a two-dimensional torus. We find situations under which this system contains a stable, elliptic periodic orbit and hence is not ergodic. This result is in contrast to the case of hard spheres interacting via inelastic collision, which are always ergodic for two particles, are conjectured to be ergodic for arbitrarily many particles, and can never contain elliptic periodic orbits.     

On the radius of convergence of the laplace series for the “Internal” potential of a gravitating torus

A correction is derived to the expression for the radius of convergence of the Laplace series for the “internal” potential of a homogenous circular torus. A spherical shell is found inside which the problem of expansion of the torus potential into a power series should be specially solved.     

Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation

Numerical studies of higher-dimensional piecewise-smooth systems have recently shown how a torus can arise from a periodic cycle through a special type of border-collision bifurcation. The present article investigates this new route to quasiperiodicity in the two-dimensional piecewise-linear normal form map. We have obtained the chart of the dynamical modes for this map and showed that border-collision bifurcations can lead to the birth of a stable closed invariant curve associated with quasiperiodic or periodic dynamics. In the parameter regions leading to the existence of an invariant closed curve, there may be transitions between an ergodic torus and a resonance torus, but the mechanism of creation for the resonance tongues is distinctly different from that observed in smooth maps. The transition from a stable focus point to a resonance torus may lead directly to a new focus of higher periodicity, e.g., a period-5 focus. This article also contains a discussion of torus destruction via a homoclinic bifurcation in the piecewise-linear normal map. Using a dc-dc converter with two-level control as an example, we report the first experimental verification of the direct transition to quasiperiodicity through a border-collision bifurcation.     

On the topological explanation of the integer quantum hall effect

We investigate the static and dynamic Kubo Hall conductivity of a non-interacting electron system in a random potential on a torus. Considering the universal covering space of the torus the Bloch theorem is applied for rational values of the filling factor. The localisation is simulated by the assumption of bound states. The Hall conductivity at zero temperatur is shown to be topologically quantized, if the Fermi energy lies in a spectral gap or in a localisation regime. The relation to previous formulations of the topological approach to the integer quantum Hall effect (QHE) is discussed.     

Effect of enclosed fluid on the dynamic response of inflated torus

Large inflatable structures have been the subject of renewed interest for scientists/engineers in recent years due to their potential space applications such as communication antennas, solar thermal propulsion and space solar power. The major advantages of using inflatable structures in space are their extremely low-weight, on-orbit deployability and inherent low launch volume. An inflated torus is a key component of many inflated space structures such as a thin membrane reflector. In view of their importance, structural static and dynamic behavior of inflated torus need to be investigated. In order to develop a more realistic model, dynamic interaction between the enclosed fluid and the torus has been included in the present work. An appreciable decrease in the modal frequencies is observed when fluid–structure interaction is taken into account. Some additional modes are also obtained. It is concluded that fluid–structure interaction significantly affects the dynamic behavior of inflatable space structures.     

Mechanical impedance gauge based on measurement of strains on a vibrating rod

An impedance gauge based on measurement of strains at two different cross-sections of a vibrating rod is analyzed and tested. The gauge rod, which may have variable characteristic impedance, is in contact with the object at one end and is driven by a harmonic vibrator at the other end. For conical and cylindrical rods explicit relations between point impedance and measured strains are derived. For a cylindrical gauge rod of steel with length 800 mm, diameter 10 mm, and distance between strain gauges 250 mm a fair agreement was generally obtained between experimental and theoretical point impedances of cylindrical test objects in the frequency range 50 Hz to 1·7 kHz. Significant improvements in accuracy over that of the tested prototype are expected to be feasible.     

Laplace series expansion of the potential of a homogeneous circular torus

The external potential of a homogeneous circular torus is represented by a series expansion in spherical functions (Laplace series). Exact analytical formulas are derived for the coefficients of this series, which can be expressed in terms of Legendre polynomials depending only on the geometrical parameter of the torus. The convergence of the series is proved and the radius of convergence is determined. The resultant expressions are verified numerically.     

Elastic theory of defects in toroidal crystals

We report a comprehensive analysis of the ground-state properties of axisymmetric toroidal crystals based on the elastic theory of defects on curved substrates. The ground state is analyzed as a function of the aspect ratio of the torus, which provides a non-local measure of the underlying Gaussian curvature, and the ratio of the defect core energy to the Young modulus. Several structural features are discussed, including a spectacular example of curvature-driven amorphization in the limit of the aspect ratio approaching one. The outcome of the elastic theory is then compared with the results of a numerical study of a system of point-like particles constrained on the surface of a torus and interacting via a short-range potential. Electronic supplementary material  Supplementary material in the form of a pdf file available from the journal web page at and are accessible for authorised users.     

Hydrodynamics of a rotating torus

The hydrodynamics of a torus is important on two counts: firstly, most stiff or semiflexible ring polymers, e.g. DNA miniplasmids are modeled as a torus and secondly, it has the simplest geometry which can describe self propelled organisms (particles). In the present work, the hydrodynamics of a torus rotating about its centerline is studied. Analytical expression for the velocity of a force free rotating torus is derived. It is found that a rotating torus translates with a velocity which is proportional to its internal velocity and to the square of the slenderness ratio, epsilon, similar to most low Reynolds number swimmers. The motion of a torus along a cylindrical track is studied numerically and it is observed that a force free torus changes its direction of motion (from a propelled state (weak wall effects) to a rolling state (strong wall effects)) as the diameter of the inner circular cylinder is increased. The rolling velocity is found to depend only on epsilon when the inner cylinder diameter approaches that of the torus.     

Flip effect in the orientation of a magnet levitating over a superconducting torus in the Meissner state

The torque between a permanent magnet and a toroidal superconductor in the Meissner state is calculated using a model previously proposed based on London’s and Maxwell’s equations. A flip effect on the stable orientation of the magnet as a function of position is demonstrated. At large distances the magnet tends to be perpendicular to the axis of the torus, but when you approach it, at a certain point there is a flip and it tends to be parallel to that axis while being closer than a certain limit. This effect can be easily used as a binary detector for proximity.     

Mechanics of atoms interacting with a carbon nanotorus: optimal configuration and oscillation behaviour

In this paper, we investigate a carbon nanotorus as a caged molecular structure interacting with an atom. Assuming that the atom is located along the central axis perpendicular to the torus, the interaction energy of the system is determined using the continuum approximation together with the Lennard-Jones potential. This approach avoids the intensive computational calculations that are involved in other modelling approaches. Numerical results are presented in terms of dimensionless variables. The results show that the optimal major radius of the torus has a linear relationship with its minor radius when the atom is symmetrically situated along the torus axis. When the atom is offset from this axis, the minimum energy location shifts away from the centre as the ratio of the major and minor radii exceeds the value of 0.90. Finally, the oscillatory behaviour for the carbon atom is investigated. Our findings predict a novel nano-oscillator which can produce frequencies in the gigahertz range.     

A semi-log-periodic array of spheroidal nanoelements: broadbanding nanoantennas

Nanoantenna arrays can offer unique possibilities for large local-field enhancement, high directivity, and also wavelength tunability over a wide spectral range. In this paper, a semi-log-periodic array of nanospheroidal elements has been studied, which exhibits a narrower beam and also a higher electric field enhancement in comparison with an equal size, equally distanced array of the same element. Through the numerical simulations, it has been presented that the suggested array is of a great potential for improvement of the directivity as well as the electric field enhancement over similar designs. Moreover, a considerable change has been observed in the electric field through gradual transformation of elements from sphere to very thin rod. Next, the effect of geometrical parameters on local-field enhancement has been investigated as a complementary job. Finally, it has been demonstrated that adjusting the number of elements can provide a narrower beam.