Chaotic behavior in lemon-shaped billiards with elliptical and hyperbolic boundary arcs

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.