ABSTRACT
Chaotic properties of a new family, ellipse hyperbola billiards (EHB), of lemon-shaped two-dimensional billiards, interpolating between the square and the circle, whose boundaries consist of hyperbolic, parabolic, or elliptical segments, depending on the shape parameter delta, are investigated classically and quantally. Classical chaotic fraction is calculated and compared with the quantal level density fluctuation measures obtained by fitting the calculated level spacing sequences with the Brody, Berry-Robnik, and Berry-Robnik-Brody distributions. Stability of selected classical orbits is investigated, and for some special hyperbolic points in the Poincaré sections, the "blinking island" phenomenon is observed. Results for the EHB billiards are compared with the properties of the family of generalized power-law lemon-shaped billiards.
ABSTRACT
Chaotic properties of the one-parameter family of oval billiards with parabolic boundaries are investigated. Classical dynamics of such billiard is mixed and depends sensitively on the value of the shape parameter. Deviation matrices of some low period orbits are analyzed. Special attention is paid to the stability of orbits bouncing at the singular joining points of the parabolic arcs, where the boundary curvature is discontinuous. The existence of such orbits is connected with the segmentation of the phase space into two or more chaotic components. The obtained results are illustrated by numerical calculations of the Poincaré sections and compared with the properties of the elliptical stadium billiards.