A stiction oscillator under slowly varying forcing: Uncovering small scale phenomena using blowup

TL;DR

This paper analyzes a mass-spring-friction oscillator with regularized stiction, revealing complex bifurcation phenomena, chaos, and small-scale effects using geometric singular perturbation theory and blowup techniques.

Contribution

It provides a geometric description of bifurcations and chaos in a stiction oscillator, resolving open problems from prior research with novel analytical methods.

Findings

Bifurcation of stick-slip limit cycles characterized

Existence of chaos proved in the oscillator system

Identification of canard-based horseshoe dynamics

Abstract

In this paper, we consider a mass-spring-friction oscillator with the friction modelled by a regularized stiction model in the limit where the ratio of the natural spring frequency and the forcing frequency is on the same order of magnitude as the scale associated with the regularized stiction model. The motivation for studying this situation comes from \cite{bossolini2017b} which demonstrated new friction phenomena in this regime. The results of Bossolini et al 2017 led to some open problems, that we resolve in this paper. In particular, using GSPT and blowup we provide a simple geometric description of the bifurcation of stick-slip limit cycles through a combination of a canard and a global return mechanism. We also show that this combination leads to a canard-based horseshoe and are therefore able to prove existence of chaos in this fundamental oscillator system.