Through the Looking-Glass of the Grazing Bifurcation: Part I - Theoretical framework

TL;DR

This paper explores how unstable periodic motions near grazing impacts in vibro-impact systems can lead to chaos, using symbolic dynamics, and examines the robustness of this phenomenon in soft impact models.

Contribution

It introduces a theoretical framework linking grazing bifurcations to chaos and demonstrates the phenomenon's robustness in soft impact models.

Findings

Chaotic behavior arises near grazing bifurcations.

Unstable periodic motions can lead to complex dynamics.

Chaos persists in soft impact models.

Abstract

It is well-known for vibro-impact systems that the existence of a periodic solution with a low-velocity impact (so-called grazing) may yield complex behavior of the solutions. In this paper we show that unstable periodic motions which pass near the delimiter without touching it may give birth to chaotic behavior of nearby solutions. We demonstrate that the number of impacts over a period of forcing varies in a small neighborhood of such periodic motions. This allows us to use the technique of symbolic dynamics. It is shown that chaos may be observed in a two-sided neighborhood of grazing and this bifurcation manifests at least two distinct ways to a complex behavior. In the second part of the paper we study the robustness of this phenomenon. Particularly, we show that the same effect can be observed in "soft" models of impacts.