Chaos in composite billiards
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AL, NONLINEAR, AND SOFT MATTER PHYSICS
Chaos in Composite Billiards V. G. Baryakhtara, V. V. Yanovskyb, S. V. Naydenovb, and A. V. Kurilob a Institute
b Institute
of Magnetism, National Academy of Sciences of Ukraine, Kiev, 03142 Ukraine for Single Crystals, National Academy of Sciences of Ukraine, ul. Lenina 60, Kharkov, 61001 Ukraine e-mail: [email protected] Received March 14, 2006
Abstract—The mechanisms and features of the chaotic behavior in billiards with ray splitting (refraction) are considered. In contrast to ordinary billiards, the law of motion in composite billiards that is coded with a sequence of ray visits to different media is shown to be deterministically chaotic. The analysis is performed in terms of a geometrical–dynamical approach in which a symmetric phase space is used instead of the ordinary Hamiltonian phase space. The chaotization elements in composite billiards of a general position are studied. The dynamics of rays in ring billiards consisting of two concentric media with different refractive indices is considered. PACS numbers: 05.45.–a DOI: 10.1134/S1063776106080127
1. INTRODUCTION Billiards belongs to an important class of dynamical systems [1, 2]. Such physical systems are generated by the free motion of material particles (or geometrical rays) inside a region with the simplest law of mirror reflection from its boundaries. The concept of deterministic chaos in classical mechanics was inextricably connected with billiards [3]. Moreover, billiards proved to be a useful theoretical model for developing the concept of quantum chaos [4]. Progress in understanding the ergodicity of statistical physics is connected with the studies of billiards [5]. Numerous physical applications of billiards are also well known. Among the best known applications are acoustic [6, 7], optical [8], and microwave [9] resonators; quantum dots [10] and superconducting Andreev billiards [11]; a Lorentz gas and a gas of colliding spheres as popular models of kinetic theory; nucleons in the short-range potential of an atomic nucleus and magnetooptic traps of cold atoms; and light collection in scintillation detectors [12]. Fundamental progress in understanding the dynamical properties of billiards (ergodicity, local instability [3], mixing [13, 14], relationship to the problem of membrane eigenmodes [15], chaos in higher dimensions [16], etc.) has been made relatively recently, not counting the extensive mathematical studies of billiards in polygons [17]. Completely chaotic, scattering Sinai billiards [13] with a negative curvature of the boundary and focusing Bunimovich stadium-type billiards [14] with a nonnegative curvature have acquired the greatest popularity among the known billiards. In this paper, we study a new type of billiards, composite billiards. A scintillation detector that consists of
two materials with different refractive indices can serve as a physical example of a system that leads to this type. This system naturally leads to a generalized model of billiards with two new elements: the
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