Related papers: Chaotic properties for billiards in circular polyg…
We construct semi-infinite billiard domains which reverse the direction of most incoming particles. We prove that almost all particles will leave the open billiard domain after a finite number of reflections. Moreover, with high probability…
Chaotic properties of symmetrical two-dimensional stadium-like billiards with elliptical arcs are studied numerically and analytically. For the two-parameter truncated elliptical billiard the existence and linear stability of several…
We consider the free motion of a point particle inside a circular billiard with periodically moving boundary, with the assumption that the collisions of the particle with the boundary are elastic so that the energy of the particle is not…
We investigate chaotic scattering on an attractive step potential with a quadrupolar deformation. The phase space features of the bound billiard are studied by using the notion of symmetry lines to find periodic orbits. We show that the…
Let $q \ge 3$ be a period. There are at least two $(1,q)$-periodic trajectories inside any smooth strictly convex billiard table, and all of them have the same length when the table is an ellipse or a circle. We quantify the chaotic…
We show that there exists a $C^2$ open dense set of convex bodies with smooth boundary whose billiard map exhibits a non-trivial hyperbolic basic set. As a consequence billiards in generic convex bodies have positive topological entropy and…
We study stochastic billiards in infinite planar domains with curvilinear boundaries: that is, piecewise deterministic motion with randomness introduced via random reflections at the domain boundary. Physical motivation for the process…
A class of non-compact billiards is introduced, namely the infinite step billiards, i.e., systems of a point particle moving freely in the domain $\Omega = \bigcup_{n\in\N} [n,n+1] \times [0,p_n]$, with elastic reflections on the boundary;…
We show that two-dimensional billiard systems are Turing complete, in the sense that the halting of any Turing machine with a given input is equivalent to a certain bounded trajectory in this system entering a specified open set. Billiards…
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…
We discuss the interplay between the piece-line regular and vertex-angle singular boundary effects, related to integrability and chaotic features in rational polygonal billiards. The approach to controversial issue of regular and irregular…
We prove the presence of topological chaos at high internal energies for a new class of mechanical refraction billiards coming from Celestial Mechanics. Given a smooth closed domain $D\in\mathbb{R}^2$, a central mass generates a Keplerian…
We describe conditions under which higher-dimensional billiard models in bounded, convex regions are fully chaotic, generalizing the Bunimovich stadium to dimensions above two. An example is a three-dimensional stadium bounded by a cylinder…
Given a domain or, more generally, a Riemannian manifold with boundary, a billiard is the motion of a particle when the field of force is absent. Trajectories of such a motion are geodesics inside the domain; and the particle reflects from…
We show that a generic analytic strongly convex billiard is "maximally chaotic" in the sense that, for every rational number $\frac{p}{q} \in \mathbb{Q} \cap (0,1)$, all intersections between the stable and unstable manifolds of maximizing…
The billiard motion inside an ellipsoid $Q \subset \Rset^{n+1}$ is completely integrable. Its phase space is a symplectic manifold of dimension $2n$, which is mostly foliated with Liouville tori of dimension $n$. The motion on each…
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…
A comprehensive study of periodic trajectories of billiards within ellipsoids in $d$-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between periodic billiard trajectories and…
We investigate the large scale chaotic, topological structure of the trajectories of an infinite sequence of dispersing, hence ergodic, $2D$ billiards with the configuration space $Q_n=\mathbb{T}^2 \setminus \bigcup_{i=0}^{n-1} D_i$, where…
In this work, we perform a statistical study on Dirac Billiards in the extreme quantum limit (a single open channel on the leads). Our numerical analysis uses a large ensemble of random matrices and demonstrates the preponderant role of…