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.

We study the quantum and classical scattering of Hamiltonian systems whose chaotic saddle is described by binary or ternary horseshoes. We are interested in parameters of the system for which a stable island, associated with the inner fundamental periodic orbit of the system exists and is large, but chaos around this island is well developed. In this situation, in classical systems, decay from the interaction region is algebraic, while in quantum systems it is exponential due to tunneling. In both cases, the most surprising effect is a periodic response to an incoming wave packet. The period of this self-pulsing effect or scattering echoes coincides with the mean period, by which the scattering trajectories rotate around the stable orbit. This period of rotation is directly related to the development stage of the underlying horseshoe. Therefore the predicted echoes will provide experimental access to topological information. We numerically test these results in kicked one dimensional models and in open billiards.

As the theory of chaotic scattering in high-dimensional systems is poorly developed, it is very difficult to determine initial conditions for which interesting scattering events, such as long delay times, occur. We propose to use symmetry breaking as a way to gain the insight necessary to determine low-dimensional subspaces of initial conditions in which we can find such events easily. We study numerically the planar scattering off a disk moving on an elliptic Kepler orbit, as a simplified model of the elliptic restricted three-body problem. When the motion of the disk is circular, the system has an integral of motion, the Jacobi integral, which is no longer conserved for nonvanishing eccentricity. In the latter case, the system has an effective five-dimensional phase space and is therefore not amenable for study with the usual methods. Using the symmetric problem as a starting point we define an appropriate two-dimensional subspace of initial conditions by fixing some coordinates. This subspace proves to be useful to define scattering experiments where the rich and nontrivial dynamics of the problem is illustrated. We consider in particular trajectories which take very long before escaping or are trapped by consecutive collisions with the disk.

We address the occurrence of narrow planetary rings and some of their structural properties, in particular when the rings are shepherded. We consider the problem as Hamiltonian scattering of a large number of non-interacting massless point particles in an effective potential. Using the existence of stable motion in scattering regions in this set up, we describe a mechanism in phase space for the occurrence of narrow rings and some consequences in their structure. We illustrate our approach with three examples. We find eccentric narrow rings displaying sharp edges, variable width and the appearance of distinct ring components (strands) which are spatially organized and entangled (braids). We discuss the relevance of our approach for narrow planetary rings.

We address the occurrence of narrow planetary rings under the interaction with shepherds. Our approach is based on a Hamiltonian framework of non-interacting particles where open motion (escape) takes place, and includes the quasi-periodic perturbations of the shepherd's Kepler motion with small and zero eccentricity. We concentrate in the phase-space structure and establish connections with properties like the eccentricity, sharp edges and narrowness of the ring. Within our scattering approach, the organizing centers necessary for the occurrence of the rings are stable periodic orbits, or more generally, stable tori. In the case of eccentric motion of the shepherd, the rings are narrower and display a gap which defines different components of the ring.