Near tangent dynamics in a class of Hamiltonian impact systems

TL;DR

This paper investigates the complex structure of tangency singularities in near integrable Hamiltonian impact systems, constructing a return map to analyze their properties and the dynamics within the singularity band.

Contribution

It introduces a piecewise smooth return map for near tangent tori in Hamiltonian impact systems and analyzes the structure and bounds of the associated singularity band.

Findings

Invariant curves exist away from the singularity set.

An upper bound for the singularity band width is derived.

Numerical simulations show long transients affected by the singular term coefficient.

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.