Dynamics of classical particles in oval or elliptic billiards with a dispersing mechanism

Abstract

Some dynamical properties for an oval billiard with a scatterer in its interior are studied. The dynamics consists of a classical particle colliding between an inner circle and an external boundary given by an oval, elliptical, or circle shapes, exploring for the first time some natural generalizations. The billiard is indeed a generalization of the annular billiard, which is of strong interest for understanding marginally unstable periodic orbits and their role in the boundary between regular and chaotic regions in both classical and quantum (including experimental) systems. For the oval billiard, which has a mixed phase space, the presence of an obstacle is an interesting addition. We demonstrate, with details, how to obtain the equations of the mapping, and the changes in the phase space are discussed. We study the linear stability of some fixed points and show both analytically and numerically the occurrence of direct and inverse parabolic bifurcations. Lyapunov exponents and generalized bifurcation diagrams are obtained. Moreover, histograms of the number of successive iterations for orbits that stay in a cusp are studied. These histograms are shown to be scaling invariant when changing the radius of the scatterer, and they have a power law slope around -3. The results here can be generalized to other kinds of external boundaries.

Scaling and self-similarity for the dynamics of a particle confined to an asymmetric time-dependent potential well

“…That self-similar structures are located in the interval (22) can be well understood since: (i) for e 0 > 1 + δ particles leave region I without suffering reflections at x = a/2 but (ii) e 0 = 1 − δ is the minimum energy for our dynamical system to work; therefore, η > 0 requires Eq. (22). Finally, in Fig.…”

“…As previous results in literature reported for other models, we expect scaling invariance of the quantity η (see for example Refs. [21] and [22] where scaling properties of multiple reflections of light beams inside a modulated waveguide and of particles trapped in an oval billiard were investigated).…”

Linear and nonlinear stability of periodic orbits in annular billiards

“…It is known that cusps can be a source of singularities in billiards 21 . At the present time, cusps created by one focusing and one dispering boundaries have not received much attention in the literature except in the recent work 18 . Prior to that publication, studies were limited to the situation with two dispersing or one dispersing and one flat wall 22 , 21 , 23 .…”

“…Recently, the related work of Correia and Zhang 17 demonstrated the existence of stability of some periodic orbits in so-called moon billiards, and ergodicity of certain other tables in that class. Linear stability and bifurcations of some periodic orbits in oval and elliptic billiards with an inner scatterer were investigated by da Costa et al 18 . Marginally unstable periodic orbits and relation to quantum chaos has been investigated by Altmann et al 19 .…”

“…There is a brief numerical investigation in 18 into the scaling of the number of collisions for excursions into such a cusp, but as yet no published rigorous results.…”