C Jung

Cantor set structures in the singularities of classical potential scattering

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. The authors 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.

Poincare map for scattering states

For scattering states in classical mechanics a map is constructed, which is an analogue to the Poincare map for bound state trajectories. Iteration of this map indicates which parts of the phase space are filled by unstable trajectories that are sensitive to small changes in the initial conditions, and which regions are filled by stable trajectories. Examples for this map are given by numerical calculations for scattering off a simple two-dimensional model potential.

Regular and irregular potential scattering

The authors present evidence for irregular behaviour in potential scattering and discuss a possible explanation in terms of unstable periodic orbits.

Scaling properties of a scattering system with an incomplete horseshoe

We investigate a parameter-dependent map describing a chaotic scattering system. In parameter ranges leading to an incomplete horseshoe we construct an approximate symbolic dynamics which describes quite well the hyperbolic component of the invariant set. Its grammatical rules are correlated with the convergence properties of the thermodynamical formalism for the measures characterizing the invariant set.

Chaotic scattering off the magnetic dipole

Classical scattering with singularities of Cantor-set type can be observed if unstable localised orbits exist whose homoclinic structures are transported to infinity by the Hamiltonian flow. An electron moving in the field of a magnetic dipole is a simple example of physical relevance to demonstrate this transport mechanism. In the asymptotic plane spanned by impact parameter and incoming direction the deflection function is singular for initial conditions leading to captured orbits. Using the method of Poincare sections, the authors find a correspondence between this set of singularities and the stable manifolds of localised orbits. The scattering data which are measured in the asymptotic region of free motion provide information about chaotic motion in a finite part of the position space.

Classical cross section for chaotic potential scattering

The authors investigate the differential cross section for a scattering system for which the existence of topological chaos in the phase space has already been shown in a previous paper. The most important result is the arrangement of an infinity of rainbow singularities into a fractal structure with a binary organisation. Its scaling behaviour is given by the eigenvalues of some periodic orbits. They discuss to what extent these results are typical for any chaotic scattering system.

Hamiltonian scattering chaos in a hydrodynamical system

The dynamics of chaotic scattering in Hamiltonian phase space can be visualized by two-dimensional open hydrodynamical systems with velocity fields which are periodic in time. Passive marker particles in the fluid trace out complicated trajectories, which are caused by the vortex structure of the fluid, i.e. the authors encounter a case of Lagrangian turbulence. By the examination of a particular model they show the applicability in hydrodynamics of ideas and methods which have been useful before in the investigation of systems describing chaotic particle scattering. In particular they show the existence of a chaotic saddle, show its influence on scattering trajectories and give some quantitative measures for it.

Hierarchical structure in the chaotic scattering off a magnetic dipole

We use the scattering off a magnetic dipole as an example of how a useful hierarchical order can be found in the set of singularities of an appropriate scattering function. This function is an unusual type of time delay function. Knowledge of the hierachical order can be used to construct a symbolic description of the dynamics. Here it is useful to have a number of symbol values which are larger than the number of basic periodic orbits of period one.

Classical chaotic scattering-periodic orbits, symmetries, multifractal invariant sets

The infinite set of periodic orbits of a chaotic system is investigated. Their globally exact symbolic dynamics is related to Birkhoffs method of iterated symmetry lines which produces symmetric periodic orbits of various classes. A complete picture emerges as to what part of the periodic orbits can be obtained by means of the Birkhoff lines. In addition, the thermodynamic formalism is applied to the multifractal structure created by the horseshoe arrangement of the periodic orbits.

Can the integrability of Hamiltonian systems be decided by the knowledge of scattering data

A method is proposed to show how scattering data of a classical Hamiltonian system can be used to decide whether the Hamiltonian function is completely integrable or not. An appropriate infinite set of scattering trajectories is linked together at infinity and the intersection of this sequence of trajectories with a surface in the set of all asymptotes is studied. The plot of these intersections provides the same information as the plots of the usual Poincare sections for bound states do. Numerical examples are given for the scattering of a spinning top, for collisional excitation of an oscillator and a rotator and for potential scattering under the additional influence of an electromagnetic field.