Near tangent dynamics in a class of Hamiltonian impact systems

Abstract

Tangencies correspond to singularities of impact systems, separating between impacting and non-impacting trajectory segments. The closure of their orbits constitute the singularity set, which, even in the simpler billiard limit, is known to have a complex structure. The properties of this set are studied in a class of near integrable two degrees-of-freedom Hamiltonian impact systems. For this class of systems, in the integrable limit, on iso-energy surfaces, tangency appears at an isolated torus. We construct a piecewise smooth iso-energy return map for the perturbed flow near such a tangent torus and study its properties. Away from the singularity set, this map has invariant curves, so, the singularity set is included in a limiting singularity band. An asymptotic upper bound of this band width is found for both non-resonant and resonant tangent tori. Numerical simulations of the dynamics inside the band reveal long transients, yet, these are made shorter when the singular term coefficient is large.