Dynamics of an impact oscillator near a degenerate graze

TL;DR

This paper provides a comprehensive geometric analysis of low-velocity impact oscillator dynamics near grazing points, including nondegenerate, degenerate, and bifurcation scenarios, revealing complex manifold structures.

Contribution

It introduces a unified geometric framework for analyzing various grazing bifurcations in impact oscillators using singularity theory.

Findings

Characterization of nondegenerate and degenerate grazing dynamics

Description of bifurcation structures at quartic grazes

Insights into stable manifold complexity in chattering orbits

Abstract

We give a complete analysis of low-velocity dynamics close to grazing for a generic one degree of freedom impact oscillator. This includes nondegenerate (quadratic) grazing and minimally degenerate (cubic) grazing, corresponding respectively to nondegenerate and degenerate {\em chatter}. We also describe the dynamics associated with generic one-parameter bifurcation at a more degenerate (quartic) graze, showing in particular how this gives rise to the often-observed highly convoluted structure in the stable manifolds of chattering orbits. The approach adopted is geometric, using methods from singularity theory.