Chaotic dynamics of the elliptical stadium billiard in the full parameter space

Abstract

Dynamical properties of the elliptical stadium billiard, which is a generalization of the stadium billiard and a special case of the recently introduced mushroom billiards, are investigated analytically and numerically. In dependence on two shape parameters δ and γ, this system reveals a rich interplay of integrable, mixed and fully chaotic behavior. Poincaré sections, the box counting method and the stability analysis determine the structure of the parameter space and the borders between regions with different behavior. Results confirm the existence of a large fully chaotic region surrounding the straight line δ = 1 − γ corresponding to the Bunimovich circular stadium billiard. Bifurcations due to the hour-glass and multidiamond orbits are described. For the quantal elliptical stadium billiard, statistical properties of the level spacing fluctuations are examined and compared with classical results.

Chaotic properties of the truncated elliptical billiards

“…This description of the billiard geometry and dynamics is consistent with our previous analysis of the elliptical stadium billiard (ESB) [30,33,34], which is a two-parameter generalization of the Bunimovich stadium billiard [4] and is a special case of the mushroom billiard [35,36,37].…”

“…It has been confirmed by analysis and numerical computation [7,26,27,28,33,34] that this billiard is fully chaotic (ergodic) for a sizeable but strictly limited region in the parameter space, defined by the stable two-bounce horizontal periodic orbit on one and the pantografic orbits on the other side. Our investigations of the ESB and TEB billiards confirm the suggestion by Del Magno [29] that in spite of apparently similar stadium-like shapes, these two billiards have essentially different dynamical properties.…”

“…In our previous work we analyzed several types of billiards with noncircular arcs (parabolic, hyperbolic, elliptical and generalized power-law), exhibiting mixed dynamics [30,31,32]. Next we investigated, in the full parameter space [33], the elliptical stadium billiards (ESB), first introduced by Donnay [7]. Here we extend the same type of analysis to the truncated elliptical billiards (TEB), which although similar in appearance, have different dynamical properties.…”

“…As explained in [33], the symbols θ, φ and φ ′ , respectively, denote the angles which the normal, the incoming path and the outcoming path make with the x-axis. The angle between the incoming (or outcoming) path and the normal to the boundary is β = (φ ′ − φ)/2.…”

“…The expression (7) is the basis for finding the existence criteria for particular periodic orbits [33]. In our further description we refer to the impact points T(x, y) in the first quadrant, with no loss of generality for the obtained results.…”