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

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 to-ward a more general understanding of chaotic scattering in higher dimen-sions. Despite of the strong restrictions it breaks the essential simplicity implicit in any two-dimensional time-independent scattering problem. In-troducing 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 singular-ities. 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.