A two-parameter study of the extent of chaos in a billiard system

Abstract

The billiard system of Benettin and Strelcyn [Phys. Rev. A 17, 773-785 (1978)] is generalized to a two-parameter family of different shapes. Its boundaries are composed of circular segments. The family includes the integrable limit of a circular boundary, convex boundaries of various shapes with mixed dynamics, stadiums, and a variety of nonconvex boundaries, partially with ergodic behavior. The extent of chaos has been measured in two ways: (i) in terms of phase space volume occupied by the main chaotic band; and (ii) in terms of the Lyapunov exponent of that same region. The results are represented as a kind of phase diagram of chaos. We observe complex regularities, related to the bifurcation scheme of the most prominent resonances. A detailed stability analysis of these resonances up to period six explains most of these features. The phenomenon of breathing chaos [Nonlinearity 3, 45-67 (1990)]-that is, the nonmonotonicity of the amount of chaos as a function of the parameters-observed earlier in a one-parameter study of the gravitational wedge billiard, is part of the picture, giving support to the conjecture that this is a fairly common global scenario. (c) 1996 American Institute of Physics.

Chaotic properties of the truncated elliptical billiards

“…If we denote the points with positive y by 1 and the points on the negative side by -1, the deviation matrix can be calculated from (14). The angle α needed in the calculation is given as…”

“…Here we test these limits numerically, with the help of the boxcounting method [14,33,38]. We calculate the Poincaré sections for a chosen pair of shape parameters, starting with n 1 randomly chosen sets of initial conditions and iterating each orbit for n 2 intersections with the x-axis, thus obtaining n 1 × n 2 points in the Poincaré diagram.…”

“…2 denoted by letter f. Stability is calculated with equation (9) and the matrix (14), where the angle α is given by sin α = 1/ √ 2. The chords are ρ ≡ ρ 1,1 = 2x…”

“…Rigorous investigations were concentrating on methods for producing fully chaotic billiards and on specific properties (Bernoulli, K-property, mixing and hyperbolicity) expressing differences between chaotic systems [5,6,7,8,9]. Various aspects of billiard dynamics have been extensively examined during last decades [10,11,12,13,14,15,16,17,18,19]. In recent years, properties of classical billiards and their quantum-mechanical counterparts were used to explain and improve performances of devices in microelectronics and nanotechnology, especially of optical microresonators in dielectrical and polymer lasers [20,21,22,23,24,25].…”

Ergodicity in Umbrella Billiards

“…Benettin and Strelcyn [7] looked at one-parameter oval tables and observed notable properties including bifurcation phenomenon, the coexistence of elliptic and chaotic regions, and the separation of the chaotic region into several invariant components. In [20] the ovals were generalized to a two-parameter family encompassing seven varieties, including special cases of lemon, moon, and a particular example of a class which in this paper we will designate as umbrella billiards, while [4] gives an alternate generalization of [7] to squash billiard tables, on which the elementary defocusing mechanism does not take place. In [25] symmetric lemon billiards were considered, and recently a class of asymmetric lemon-shaped convex billiard tables were constructed in [13], obtained by intersection of two disks in the plane.…”