Scattering off two oscillating disks: Dilute chaos

Abstract

We investigate the role of the unstable periodic orbits and their manifolds in the dynamics of a time-dependent two-dimensional scattering system. As a prototype we use two oscillating disks on the plane with the oscillation axes forming an angle theta. The phase space of the system is five dimensional and it possesses a variety of families of unstable periodic orbits (UPOs) with intersecting manifolds. We perform numerical experiments to probe the structure of distinct scattering functions, in one and two dimensions, near the location of the UPOs. We find that the corresponding manifolds occur only in a very particular and localized way in the high-dimensional phase space. As a consequence the underlying fractal structure is ubiquitous only in higher-dimensional, e.g., two-dimensional, scattering functions. Both two-dimensional and one-dimensional scattering functions are dominated by seemingly infinite sequences of discontinuities characterized by small values of the magnitude of the projectile's outgoing velocity. These peaks accumulate toward the phase-space locations of the UPOs, with a rate which monotonically depends on the corresponding instability exponent. They represent the intersections of the set of the initial conditions with invariant sets of larger dimensionality embedded in the phase space of the system, which are not directly related with the UPOs. We adopt the term "dilute chaos" to characterize these phenomenological aspects of the scattering dynamics.

Pumping angular momentum by driven chaotic scattering

“…Note that the sense of precession is inverted under time reversal t → −t (keeping B → B), revealing the time arrow inherent in (5). (3,4). In each of the two sectors 1 and 2, the field is isotropic, with an angle of π/2 between the two sectors.…”

“…The field configuration defined in (3,4) implies that within the intervals [−a, 0] and [0, a], the field varies only in magnitude but not in direction. This allows for further simplifications.…”

“…Yet there are numerous reasons to give them a closer look: An internal freedom may provide the additional dimension required to render a scattering system with a single external degree of freedom chaotic [1,2,3,4]. It may also act as a reservoir that absorbs energy from the external motion and stores it temporarily, thus giving the scattering process a transient inelastic character.…”

Scattering off an oscillating target: Basic mechanisms and their impact on cross sections

“…Recently, it has been found that complex behavior with different characteristics can be observed in scattering systems involving a time-dependent scattering potential [23,24,25,26,27,28,29]. As a representative example of such a system the scattering of freely moving, non-interacting particles off two hard, infinitely heavy, oscillating discs on the plane has been studied.…”

“…[25] the term dilute chaos was introduced to describe the behavior characterized by the accumulation of peaks, associated with processes leading to maximum energy exchange between scattered particles and an oscillating target around the position of the UPOs of the system. The set of the observed peaks possesses a self-similar structure either in an approximative sense occurring only between two scales determined by the geometry of the particular setup [24] or in a local sense in the neighborhood of specific isolated points in phase space [25]. The significant role of processes with maximum energy exchange in the scattering off a time-dependent hard potential becomes more transparent in the case of scattering off a single oscillating disc [26].…”