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 constitutes 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. 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. In the Diophantine case the band extent to both the impacting and nonimpacting regimes is, asymptotically, of order (\epsilon). In the resonance case its asymptotic extent to the nonimpacting regime is at most of order (\sqrt\epsilon) whereas its maximal asymptotic extent to the impacting regime is of order (\epsilon^2/3). Studying numerically a truncated standard form of the return map which is integrable at (\epsilon=0) and has a square root singularity with a singularity parameter (\alpha) reveals that only the strongest resonance obeys this maximal extent resonance scaling. We find that for all but one resonant case the singularity band has connecting orbits from its lower to its upper boundaries. The number of iterates required for an orbit to visit close to both the lower and upper boundaries of the singularity band diverges as (\epsilon\alpha^2) is decreased, giving rise to a multitude of transient complex phenomena. On the other hand, large (\alpha) values at a fixed small (\epsilon) lead to a significant reduction in this connection time.