Topology of Vibro-Impact Systems in the Neighborhood of Grazing

TL;DR

This paper investigates the topological structure of vibro-impact systems near grazing bifurcations, providing a robust mathematical model to explain the complex, sensitive attractors observed experimentally and numerically.

Contribution

It introduces a new approach to model non-hyperbolic dynamics near grazing, resolving contradictions between theoretical invisibility and observed attractors.

Findings

A new mathematical model for non-hyperbolic dynamics near grazing.

Explanation of the sensitivity and visibility of attractors in experiments.

Conditions for the existence of grazing families of periodic solutions.

Abstract

The grazing bifurcation is considered for the Newtonian model of vibro-impact systems. A brief review on the conditions, sufficient for existence of a grazing family of periodic solutions, is given. The properties of these periodic solutions are discussed. A plenty of results on the topological structure of attractors of vibro-impact systems is known. However, since the considered system is strongly nonlinear, these attractors may be invisible or, at least, very sensitive to changes of parameters of the system. On the other hand, they are observed in experiments and numerical simulations. We offer (Theorem 2) an approach which allows to explain this contradiction and give a new robust mathematical model of the non-hyperbolic dynamics in the neighborhood of grazing.