Reconstruction of the chaotic set from classical cross section data

We study a paradigm of a chaotic scattering system with one open and one closed degree of freedom. We show how the development scenario of the chaotic invariant set is reflected in the changes of the rainbow singularities in the classical cross section. This allows us to reconstruct the actual development stage of the chaotic set from the knowledge of the cross section. We do this by following the rainbow structure of the cross section along a curve in parameter space which connects the case we are interested in with a known case, for example, the case of a complete horseshoe. We conclude with a discussion about the generality of our procedure and of its application to various systems.