The decay of a normally hyperbolic invariant manifold to dust in a three degrees of freedom scattering system

Abstract

We study a prototypical three degrees of freedom chaotic scattering system depending on a perturbation parameter. For one limiting case of the parameter, the system has a conserved quantity and can be reduced to a two degrees of freedom system. And for the other limiting case, the chaotic invariant set is completely hyperbolic and forms a dust. For the reducible case the dynamics of the Poincaré map is directed by a normally hyperbolic invariant manifold and this four dimensional map can be understood with the help of a Cartesian product of a circle with a continuous stack of reduced two dimensional maps. As the perturbation sets in, for small values of the perturbation parameter the normally hyperbolic invariant manifold persists and only for larger values of the perturbation it turns into dust. We give details of this decay and describe the corresponding changes of singularities in the scattering functions. An important tool for the study of the development scenario of the normally hyperbolic invariant manifold is the restriction of the Poincaré map to this subset itself. It behaves like a two dimensional perturbed twist map. We explain how it can be approximated numerically by a combination of the full dimensional Poincaré map with a projection.