Stability in High Dimensional Steep Repelling Potentials

Abstract

ABSTRACT. The appearance of elliptic periodic orbits in families of n-dimensional smooth repelling billiard-like potentials that are arbitrarily steep is established for any finite n. Furthermore, the stability regions in the parameter space scale as a power-law in 1/n and in the steepness parameter. Thus, it is shown that even though these systems have a uniformly hyperbolic (albeit singular) limit, the ergodicity properties of this limit system are destroyed in the more realistic smooth setting. The considered example is highly symmetric and is not directly linked to the smooth many particle problem. Nonetheless, the possibility of explicitly constructing stable motion in smooth n degrees of freedom systems limiting to strictly dispersing billiards is now established.

On topological entropy of billiard tables with small inner scatterers

“…For the now so-called Sinai billiard system, he proved that it has positive measure-theoretic (Kolmogorov-Sinai) entropy and is hyperbolic almost everywhere. See also [11,18,32,34] and also [14,26,[29][30][31] for relevant and recent results and references therein. In contrast, rays converge after reflecting from convex boundaries.…”

“…The hard scatterer case can be modelled by considering the limit lim →0 V ρ (·, ), cf. [30,31]. This limiting case describes the Sinai billiard on the two torus with a circular scatterer of diameter ρ, and in this case the lower bound goes to infinity at the rate −2 ln ρ + O(1) as ρ goes to zero.…”

“…a soft scatterer, cf. [14,30,31]. The author showed that the lower bound can be made arbitrarily large provided that ρ and are sufficiently small.…”

On Near Integrability of Some Impact Systems

“…By this approach, to better understand systems with very steep potentials at the domain's boundary, one studies the limit system in which the steep part is replaced by impacts. Once the dynamics under the HIS are known, one establishes which of its features persist [27,17] and how those which do not persist bifurcate [30,28].…”

Smooth phase transition of energy equilibration in a springy Sinai billiard

“…A partial answer is provided by the ergodic hy- * Electronic address: kushals@iiserb.ac.in pothesis which states that all accessible microstates of a given system are equiprobable over sufficiently long periods of time [4,5]. However, there are very few dynamical systems which have been actually proven to be ergodic [6,7,[9][10][11][12]. And even for ergodic systems, the time required for ergodization may be so long that it may be practically irrelevant.…”