Elliptical Billiards Research Articles

Phase-space composition of driven elliptical billiards and its impact on Fermi acceleration

We demonstrated very recently [Lenz, New J. Phys. 11, 083035 (2009)] that an ensemble of particles in the driven elliptical billiard shows a surprising crossover from subdiffusion to normal diffusion in momentum space. This crossover is not parameter induced, but rather occurs dynamically in the evolution of the ensemble. In this work, we consider three different driving modes of the elliptical billiard and perform a comprehensive analysis of the corresponding four-dimensional phase space. The composition of this phase space is different in the high-velocity regime compared to the low-velocity regime. We will show, among others, by investigating periodic orbits and probability distributions of laminar phases that the stickiness properties, which eventually determine the diffusion, are intimately connected with this change in the composition of the phase space with respect to velocity. In the course of the evolution, the accelerating ensemble thus explores regions of varying stickiness, leading to the mentioned crossover in the diffusion.

Chaotic properties of the truncated elliptical billiards

Chaotic properties of symmetrical two-dimensional stadium-like billiards with piecewise flat and elliptical segments are studied numerically and analytically. Their sensitivity to small variations of the shape parameters can be usefully applied for optimal construction of the dielectric and polymer optical microresonators. For the two-parameter truncated elliptical billiards (TEB) the existence and linear stability of several periodic orbits are investigated in the full parameter space. Poincaré plots are computed and used for evaluation of the chaotic fraction of the phase space by means of the box-counting method. A highly chaotic behavior prevails in the region of elongated elliptical arcs, where most of the existing orbits are either neutral or unstable. In the parameter space, the upper limit of the fully chaotic behavior is reached when the relation between the two shape parameters becomes δ = 1 - γ 2 , corresponding to truncated circles. Above this limit, for flattened elliptical arcs, mixed dynamics with numerous stable elliptic islands is found. In both regions parabolic orbits are present, many of them identical to orbits within an ellipse. These properties of the TEB differ remarkably from the behavior in the elliptical stadium billiards (ESB), where the chaotic region in the parameter space was strictly bounded from both sides. In order to follow the transition from TEB to ESB, a generalisation to a novel three-parameter family (GTEB) of stadium-like boundary shapes with elliptical arcs is proposed.

A curvature theory for discrete surfaces based on mesh parallelity

We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the faces’ areas and mixed areas. Remarkably these notions are capable of unifying notable previously defined classes of surfaces, such as discrete isothermic minimal surfaces and surfaces of constant mean curvature. We discuss various types of natural Gauss images, the existence of principal curvatures, constant curvature surfaces, Christoffel duality, Koenigs nets, contact element nets, s-isothermic nets, and interesting special cases such as discrete Delaunay surfaces derived from elliptic billiards.

Bifurcations of Liouville tori in elliptical billiards

A detailed description of topology of integrable billiard systems is given. For elliptical billiards and geodesic billiards on ellipsoid, the corresponding Fomenko graphs are constructed.

Evolutionary phase space in driven elliptical billiards

We perform the first long-time exploration of the classical dynamics of a driven billiard with a four-dimensional phase space. With increasing velocity of the ensemble, we observe an evolution from a large chaotic sea with stickiness due to regular islands to thin chaotic channels with diffusive motion leading to Fermi acceleration. As a surprising consequence, we encounter a crossover, which is not parameter induced but rather occurs dynamically, from amplitude-dependent tunable subdiffusion to universal normal diffusion in momentum space.

Trajectory tracking for a particle in elliptical billiards

An infinitely rigid unitary mass (particle) is considered, moving on a planar region delimited by a rigid elliptical barrier (elliptical billiards) under the action of proper control forces. A class of periodic trajectories, involving an infinite sequence of non-smooth impacts between the mass and the barrier at fixed times, is found by using an LMIs based procedure. The jumps in the velocities at the impact times render difficult (if not impossible) to obtain the classical stability and attractivity properties for the dynamic system describing the tracking error behaviour. Hence, the tracking control problem is properly stated using notions similar to the quasi stability concept in V. Lakshmikantham, D.D. Bainov and P.S. Simeonov, Theory of Impulsive Differential Equations, 6, World Scientific, 1989. A controller (whose state is subject to discontinuities) based on the internal model principle is shown to solve the proposed tracking problem, giving rise to control forces that are piecewise continuous function of time, with discontinuities at the desired impact times and at the impact times of the particle with the barrier.

Spectra analysis of the two-dimensional elliptic billiards

The correspondence between the quantum spectra and the classical orbits of the two-dimensional elliptic billiards is investigated. In order to study systems without analytic solutions in the future, an expansion method for stationary states (EMSS) to get highly accurate numerical solutions is applied and tested. The closed-orbit theory was extended to the open orbit case. By comparing the Fourier-transformed spectra with the classical orbits, we find that the peak positions in the quantum spectra match with the lengths of the classical orbits with small numerical differences. This is another example showing that semiclassical method provides a bridge between quantum and classical mechanics.

Chaotic Properties of the Elliptical Stadium Billiard

The two-parameter family of elliptical stadium billiards is discussed. Special attention is paid to the one-parameter subfamily inscribed into the square which interpolates between the circle and the square. It is shown that the classical dynamics of such system is mixed for all values of the shape parameter. For the same system the quantal spectra and wave functions are calculated, and the classical and quantal chaotic fraction values are compared. Other types of elliptical stadium billiards are briefly discussed.

On elliptical billiards in the Lobachevsky space and associated geodesic hierarchies

We derive Cayley’s type conditions for periodical trajectories for the billiard within an ellipsoid in the Lobachevsky space. It appears that these new conditions are of the same form as those obtained before for the Euclidean case. We explain this coincidence by using theory of geodesically equivalent metrics and show that Lobachevsky and Euclidean elliptic billiards can be naturally considered as a part of a hierarchy of integrable elliptical billiards.

Chaotic dynamics and orbit stability in the parabolic oval billiard.

Chaotic properties of the one-parameter family of oval billiards with parabolic boundaries are investigated. Classical dynamics of such billiard is mixed and depends sensitively on the value of the shape parameter. Deviation matrices of some low period orbits are analyzed. Special attention is paid to the stability of orbits bouncing at the singular joining points of the parabolic arcs, where the boundary curvature is discontinuous. The existence of such orbits is connected with the segmentation of the phase space into two or more chaotic components. The obtained results are illustrated by numerical calculations of the Poincaré sections and compared with the properties of the elliptical stadium billiards.

Dynamic control of an embedded cavity resonator

We propose two embedded elliptic billiards as a possible cavity resonator. We analyze the ray and wave optic properties of the resonators, mostly in 2D but also in 3D. We show that this system can produce special whispering gallery modes that can be controlled by changing the parameters of the billiards. Our results indicate that this configuration may work well as a short wavelength practical table-top cavity resonator.

Ergodicity of a class of truncated elliptical billiards

We consider a class of billiard tables obtained byintersecting elliptical domains x2/a2 + y2⩽1,a>1 withhorizontal strips |y|⩽h<1. The boundary of these tables consists oftwo elliptical arcs connected by two parallel straight segments. We provethat the billiards in these tables have non-vanishing Lyapunov exponentsfor h<min (1/a,1/(2)1/2), and are ergodic for h<1/(1 + a2)1/2.

Perturbations of elliptic billiards

The dynamical system arising from the problem of billiards is a classical example where the theory of twist maps can be applied. In the case of an elliptic billiard table, the corresponding twist map is integrable and has a saddle connection between two hyperbolic period two points. Using a discrete analog to the Melnikov method, we are able to show that this saddle connection can be deformed into a transversal heteroclinic connection under certain analytic perturbations of the table. From the formulas that we get, we can show that the splitting of the separatrices is exponentially small as a function of the eccentricity of the original unperturbed elliptic table. In addition, we also include a characterization of the period two periodic points for any billiard table.

Noncoincidence of geodesic lengths and hearing elliptic quantum billiards

Assume that the planar region Ω has aC1 boundary ∂Ω and is strictly convex in the sense that the tangent angle determines a point on the boundary. The lengths of invariant circles for the billiard ball map (or caustics) accumulate on |∂Ω|. It follows from direct calculations and from relations between the lengths of invariant circles and the lengths of trajectories of the billiard ball map that under mild assumptions on the lengths of some geodesics the region satisfies the strong noncoincidence condition. This condition plays a role in recovering the lengths of closed geodesics from the spectrum of the Laplacian. Asymptotics for the lengths of invariant circles and an application to ellipses are discussed. In addition; some examples regarding strong non coincidence are given.

Poincaré - Melnikov - Arnold method for analytic planar maps

The Poincaré - Melnikov - Arnold method for planar maps gives rise to a Melnikov function defined by an infinite and ( a priori) analytically uncomputable sum. Under an assumption of meromorphicity, residues theory can be applied to provide an equivalent finite sum. Moreover, the Melnikov function turns out to be an elliptic function and a general criterion about non-integrability is provided. Several examples are presented with explicit estimates of the splitting angle. In particular, the non-integrability of non-trivial symmetric entire perturbations of elliptic billiards is proved, as well as the non-integrability of standard-like maps. 33E05

On the link between umbilic geodesics and soliton solutions of nonlinear PDEs

In this paper we describe a new class of soliton solutions, called umbilic solitons, for certain nonlinear integrable PDEs. These umbilic solitons have the property that as the space variable x tends to infinity, the solution tends to a periodic wave, and as x tends to minus infinity, it tends to a phase shifted wave of the same shape. The equations admitting solutions in this new class include the Dym equation and equations in its hierarchy. The methods used to find and analyse these solutions are those of algebraic and complex geometry. We look for classes of solutions by constructing associated finite-dimensional integrable Hamiltonian systems on Riemann surfaces. In particular, in this setting we use geodesics on n -dimensional quadrics to find the spatial, or x -flow, which, together with the commuting t -flow given by the equation itself, defines new classes of solutions. Amongst these geodesics, particularly interesting ones are the umbilic geodesics, which then generate the class of umbilic soliton solutions. This same setting also enables us to introduce another class of solutions of Dym-like equations, which are related to elliptic and umbilic billiards.

Elliptical billiards and hyperelliptic functions

The geometrical properties of the elliptical billiard system are related to Poncelet’s theorem. This theorem states that if a polygon is inscribed in a conic and circumscribed about a second conic, every point of the former conic is a vertex of a polygon with the same number of sides and the same perimeter. Chang and Friedberg have extended this theorem to three and higher dimensions. They have shown that the geometrical properties of the hyperelliptic billiard system are related to the algebraic character of a Poincaré map in the phase space. The geometrical and algebraic properties of the system can be understood in terms of the analytical structure of the equations of motion. These equations form a complete system of Abelian integrals. The integrability of the physical system is reflected by the topology of the Riemann surfaces associated to these integrals. The algebraic properties are connected with the existence of addition formulas for hyperelliptic functions. The main purpose of this study is to establish such a connection, and to provide an algebraic proof of Poncelet’s theorem in three and higher dimensions.

A new integrable gravitational billiard

We discuss gravitational billiards, i.e. the two-dimensional motion of a paint mass inside a hard boundary curve under the influence of a constant (e.g. gravitational) field. A parabolic boundary is shown to be a new example of integrable billiards. The system has a second integral of motion in addition to the energy, which is constructed analytically. Stability and bifurcation properties ofthe central periodic orbits arc discussed. the results also shed new light on the known integrable care of elliptic non-gravitational billiards.

Elliptical billiards and Poncelet’s theorem

Two- and three-dimensional billiard systems with elliptical and ellipsoidal boundaries, respectively, are studied. It is known that the trajectory of such a two-dimensional system generates a caustic conic curve. Many properties of the elliptical billiard system in the language of projective geometry can be described. One of these properties is that if a trajectory is closed after p bounces, then all trajectories sharing the same caustic conic are also closed after p bounces. In projective geometry, this is known as Poncelet’s theorem. Many of the two-dimensional results are extended to three dimensions. In particular, it is shown that all straight segments of a trajectory in a three-dimensional ellipsoidal billiard system are always tangent to two caustic quadrics. If a trajectory is closed after p bounces, then all trajectories sharing the same two caustic quadrics are also closed after p bounces. A generalized Poncelet’s theorem in three dimensions is thus established. On the basis of numerical studies, it is conjectured that this generalized Poncelet’s theorem is also valid for an arbitrary finite number field. The implications of the results are discussed and their extension to n dimensions is outlined.