Related papers: Coding of billiards in hyperbolic 3-space
Symbolic dynamics for homoclinic orbits in the two-dimensional symmetric map, $x_{n+1}+cx_{n}+x_{n-1}=3x_{n}^3$, is discussed. Above a critical $c^{\ast}$, the system exhibits a fully-developed horse-shoe so that its global behavior is…
We introduce a new class of billiard systems in the plane, with boundaries formed by finitely many arcs of confocal conics such that they contain some reflex angles. Fundamental dynamical, topological, geometric, and arithmetic properties…
A billiard is a dynamical system in which a particle alternates between motion in a straight line and specular reflection from a boundary. For billiards in non-convex areas bounded by segments of confocal quadrics are studied. The topology…
Dynamical properties of the elliptical stadium billiard, which is a generalization of the stadium billiard and a special case of the recently introduced mushroom billiards, are investigated analytically and numerically. In dependence on two…
We define billiards in the context of sub-Finsler Geometry. We provide symplectic and variational (or rather, control theoretical) descriptions of the problem and show that they coincide. We then discuss several phenomena in this setting,…
We formulate a conjecture for the second generation characters of indecomposable tilting modules for ${\rm SL}_3$. This gives many new conjectural decomposition numbers for symmetric groups. Our conjecture can be interpreted as saying that…
We investigate the classical scattering dynamics of the driven elliptical billiard. Two fundamental scattering mechanisms are identified and employed to understand the rich behavior of the escape rate. A long-time algebraic decay which can…
Sufficiently accurate finite state models, also called symbolic models or discrete abstractions, allow one to apply fully automated methods, originally developed for purely discrete systems, to formally reason about continuous and hybrid…
Dynamical focusing of ensembles of neutral particles in energy and configuration space has been demonstrated recently [C. Petri et al. 2010, Phys. Rev. E (R) {\bf 82}, 035204] using time-dependent elliptical billiards. The interplay of…
We consider classical dynamical properties of a particle in a constant gravitational force and making specular reflections with circular, elliptic or oval boundaries. The model and collision map are described and a detailed study of the…
This article is devoted to a study of the asymptotic dynamics of generic solutions of the Einstein vacuum equations toward a generic spacelike singularity. Starting from fundamental assumptions about the nature of generic spacelike…
The dynamics in three-dimensional billiards leads, using a Poincar\'e section, to a four-dimensional map which is challenging to visualize. By means of the recently introduced 3D phase-space slices an intuitive representation of the…
We derive an analytical trace formula for the level density of the two-dimensional elliptic billiard using an improved stationary phase method. The result is a continuous function of the deformation parameter (eccentricity) through all…
We introduce the notion of a Billiard Array. This is an equilateral triangular array of one-dimensional subspaces of a vector space $V$, subject to several conditions that specify which sums are direct. We show that the Billiard Arrays on…
The degenerate Lagrangian system describing a lot of cosmological models is considered. When certain restrictions on the parameters of the model are imposed, the dynamics of the model near the "singularity" is reduced to a billiard on the…
We present a computational scheme based on classical molecular dynamics to study chaotic billiards in static external magnetic fields. The method allows to treat arbitrary geometries and several interacting particles. We test the scheme for…
We give lower bound on the number of periodic billiard trajectories inside a generic smooth strictly convex closed surface in 3-space: for odd n, there are at least 2(n-1) such trajectories. We apply a topological approach based on the…
It is demonstrated numerically that smooth three degrees of freedom Hamiltonian systems which are arbitrarily close to three dimensional strictly dispersing billiards (Sinai billiards) have islands of effective stability, and hence are…
We study dynamics in a neighborhood of a nonhyperbolic fixed point or an irreducible homoclinic tangent point. General type conditions for the existence of infinite sets of periodic points are obtained. A new method, based on the study of…
The goal of this paper is an analysis of the geometry of billiards in ellipses, based on properties of confocal central conics. The extended sides of the billiards meet at points which are located on confocal ellipses and hyperbolas. They…