Hard chaos in magnetic billiards (on the hyperbolic plane)

Abstract

Dynamics of a gravitational billiard with a hyperbolic lower boundary Chaos 9, 841 (1999);In this paper some results on the local and global stability analysis of magnetic billiard systems, established on two dimensional Riemannian manifolds of constant curvature are presented, with particular emphasis on the hyperbolic plane. For special billiards, possessing a discrete group of ͑rotational or translational͒ symmetry, a geometrical theorem, illustrated by numerical simulations, is given on the stability of trajectories with the same symmetry. We also present sufficient criteria for the global hyperbolicity of the dynamics ͑hard chaos͒, and give lower estimations for the Lyapunov exponent in terms of the shape of the billiard.

Hyperbolic Magnetic Billiards on Surfaces¶of Constant Curvature

“…See, however, [Ta2] for some results on chaotic billiards on the hyperbolic plane, and [Vet1], [Vet2], [KSS] for some results on hyperbolic billiards on a general Riemannian surface. In our recent work [GSG] we have generalized Wojtkowski's criterion of hyperbolicity for planar billiards [Wo2] to billiards on arbitrary surfaces of constant curvature.…”

“…As it has been shown in [Ta2], [Ta3] (see also [K] for the planar case) the free-flight evolution of χ satisfies the Ricatti equation:…”

“…Magnetic billiards on the plane have been considered in many works [BR], [K], [BK], [Ta1] and on the hyperbolic plane in [Ta2]. The study of such billiards is strongly motivated by mesoscopic physics, where such billiard models are used as simplified version of the mesoscopic devices in the presence of magnetic fields.…”

“…Let κ be the curvature of ∂Q at m, and let θ be the angle between v and the positive tangent vector to ∂Q at m. Then, the extension of the well known formula for planar billiards (see e.g., [Si]) to magnetic billiards on arbitrary surfaces (see [Ta2], [Ta3]) gives…”

Violation of adiabaticity in magnetic billiards due to separatrix crossings

“…7,16 There are several investigations of stochastization of motion in the magnetic billiards with and without curvature of the boundaries of domain where charged particles move. 18,23,43,44 One of possible triggers of stochastization of motion is the destruction of adiabatic invariance (for magnetic billiards, the role of this invariant is played by the classical magnetic moment 35 ). It was shown that destruction of the adiabaticity in magnetic billiards can realize in case of an inhomogeneous magnetic field.…”