Related papers: Periodic Billiards in Isosceles Triangles

We give an explicit sub-exponential estimate on the growth rate of periodic orbits and generalized diagonals for typical triangle billiards.

In this paper we present new results regarding the periodicity of outer billiards in the hyperbolic plane around polygonal tables which are tiles in regular two-piece tilings of the hyperbolic plane.

As the parameters of a piecewise-smooth system of ODEs are varied, a periodic orbit undergoes a bifurcation when it collides with a surface where the system is discontinuous. Under certain conditions this is a grazing-sliding bifurcation.…

Using Marden's Theorem from geometric theory of polynomials, we show that for every triangle there is a unique ellipse such that the triangle is a billiard trajectory within that ellipse. Since $3$-periodic trajectories of billiards within…

We introduce a new equivalence relation on the set of all polygonal billiards. We say that two billiards (or polygons) are order equivalent if each of the billiards has an orbit whose footpoints are dense in the boundary and the two…

We give the asymptotic growth of the number of primitive periodic trajectories of a two dimensional dispersive billiard, when we prescribe their number of bounces on one of the obstacles.

A right triangular billiard system is equivalent to the system of two colliding particles confined in a one-dimensional box. In spite of their seeming simplicity, no definite conclusion has been drawn so far concerning their ergodic…

The study of polygonal billiards, particularly those in the regular pentagon, has been the subject of two recent papers. One of these papers approaches the problem of discovering the periodic trajectories on the pentagon by identifying…

An annular billiard is a dynamical system in which a particle moves freely in a disk except for elastic collisions with the boundary, and also a circular scatterer in the interior of the disk. We investigate stability properties of some…

We investigated experimentally the ray-wave correspondence in organic microlasers of various triangular shapes. Triangular billiards are of interest since they are the simplest cases of polygonal billiards and the existence and properties…

This paper had a serious error. In fixing the error the emphasis of the paper has changed completely, thus meriting a new name: ``Periodic orbits in right triangles''. I have made a new submission to arXiv with this name.

Hidden attractors are present in many nonlinear dynamical systems and are not associated with equilibria, making them difficult to locate. Recent studies have demonstrated methods of locating hidden attractors, but the route to these…

Multiplicities of periodic orbit lengths for non-arithmetic Hecke triangle groups are discussed. It is demonstrated both numerically and analytically that at least for certain groups the mean multiplicity of periodic orbits with exactly the…

Can any secrets still be shed by that much studied, uniquely integrable, Elliptic Billiard? Starting by examining the family of 3-periodic trajectories and the loci of their Triangular Centers, one obtains a beautiful and variegated gallery…

We consider periodic second-order equations having an ordered pair of lower and upper solutions and show the existence of asymptotic trajectories heading towards the maximal and minimal periodic solutions which lie between them.

Billiards in ellipses have a confocal ellipse or hyperbola as caustic. The goal of this paper is to prove that for each billiard of one type there exists an isometric counterpart of the other type. Isometry means here that the lengths of…

The famous conjecture of V.Ya.Ivrii (1978) says that {\it in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero}. In the present paper we study the complex version of Ivrii's…

A trajectory of a harmonic oscillator obeying the Schreodinger wave equation is exactly derived and illustrated. The trajectory resembles well the classical orbit between the turning points, and also runs through the tunneling region. The…

In this paper we define and study the billiard problem on bounded regions on surfaces of constant curvature. We show that this problem defines a 2-dimensional conservative and reversible dynamical system, defined by a Twist diffeomorphism,…

We investigate dynamical properties of chaotic trajectories in mushroom billiards. These billiards present a well-defined simple border between a single regular region and a single chaotic component. We find that the stickiness of chaotic…