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.
Keywords
Cite
@article{arxiv.2110.11637,
title = {Near tangent dynamics in a class of Hamiltonian impact systems},
author = {M. Pnueli and V. Rom-Kedar},
journal= {arXiv preprint arXiv:2110.11637},
year = {2021}
}