Cantor set structures in the singularities of classical potential scattering

Abstract

A classical mechanical system is analysed which exhibits complicated scattering behaviour. In the set of all incoming asymptotes there is a fractal subset on which the scattering angle is singular. Though in the complement of this Cantor set the deflection function is regular, one can choose impact parameter intervals leading to arbitrarily complicated trajectories. We show how the complicated scattering behaviour is caused by unstable periodic orbits having homoclinic and heteroclinic connections. Thereby a hyperbolic invariant set is created leading to horseshoe chaos in the flow. This invariant set contains infinitely many unstable localised orbits (periodic and aperiodic ones). The stable manifolds of these orbits reach out into the asymptotic region and create the singularities of the scattering function.

Fractal basins of escape and the formation of spiral arms in a galactic potential with a bar

“…This problem is well understood from the viewpoint of chaos theory (e.g. Bleher et al 1989Bleher et al , 1990Boyd & McMillan 1992Chen et al 1990;Ding et al 1990;Eckhardt & Jung 1986;Eckhardt 1987Eckhardt , 1988Gaspard & Rice 1989;Hénon 1988;José et al 1992;Jung 1987;Jung & Scholz 1987;Jung & Pott 1989;Jung & Richter 1990;Jung & Tel 1991;Jung et al 1995Jung et al , 1999Lai et al 1993Lai et al , 2000Lau et al 1991;Motter & Lai 2002;Rückerl & Jung 1994;Seoane et al 2006Seoane et al , 2007Seoane & Sanjuán 2008;Sweet & Ott 2000) and has been applied in the astrophysical context to, e.g., the scattering off black holes (e.g. Aguirregabiria 1997;de Moura & Letelier 2000) and three-body systems (e.g.…”

The vibrational dynamics of 3D HOCl above dissociation

“…Studies of fractal scattering processes have focused on systems with two degrees of freedom (2D) because in 2D they are easy to visualize [9][10][11][12][13][14][15] and characterize using Poincaré surfaces of section (SOS). In 2D systems it is possible to follow the flow of stable and unstable manifolds, as they form an increasingly complex network of tendrils in the phase space.…”

“…This fractal set of singularities occurs when the incoming particle asymptotes intersect the tangles of the invariant manifolds in the asymptotic region. The recipe for extracting information about chaotic scattering processes and the underlying fractal structure that governs that dynamics has been developed by Jung and others for a variety of model systems [9][10][11][12][13]33]. The method described here for obtaining the underlying fractal structure for chaotic scattering process can also be applied to scattering processes in the absence of a radiation field, as has been shown in [14] where the chaotic scattering dynamics of a molecular system is analysed.…”

Nonperiodic echoes from mushroom billiard hats

“…Particularly quantum or wave realizations of such objects have become popular, since flat microwave cavities, socalled microwave billiards, are used by experimentalists at different laboratories [4,5,6,7,8,9,10], and these experiments mimic some properties of mesoscopic devices. From a mathematical point of view one of the advantages of billiards is that, in many instances, chaotic properties can be proven for cases of complete chaos [1,11,12,13] and more recently even for mixed systems [14,15].…”