The chaotic set and the cross section for chaotic scattering beyond in 3 degrees of freedom

TL;DR

This paper investigates chaotic scattering in a three-degree-of-freedom system with one open and two closed degrees, analyzing the topological structures and fractal patterns in the cross section to understand invariant sets and inverse scattering.

Contribution

It extends the understanding of chaotic scattering to higher dimensions by analyzing the topological structure and fractal patterns in the cross section with broken symmetry.

Findings

Rainbow singularities form fractal patterns in the cross section.

Structures in the cross section reflect the fractal invariant set.

Inverse scattering can infer invariant set topology from the cross section.

Abstract

This article treats chaotic scattering with three degrees of freedom, where one of them is open and the other two are closed, as a first step toward a more general understanding of chaotic scattering in higher dimensions. Despite of the strong restrictions it breaks the essential simplicity implicit in any two-dimensional time-independent scattering problem. Introducing the third degree of freedom by breaking a continuous symmetry, we first explore the topological structure of the homoclinic/heteroclinic tangle and the structures in the scattering functions. Then we work out implications of these structures for the doubly differential cross section. The most prominent structures in the cross section are rainbow singularities. They form a fractal pattern which reflects the fractal structure of the chaotic invariant set. This allows to determine structures in the cross section from the invariant set and conversely, to obtain information about the topology of the invariant set from the cross section. The latter is a contribution to the inverse scattering problem for chaotic systems.