# Canards in stiction: on solutions of a friction oscillator by regularization

## TL;DR

This paper investigates the complex dynamics of a friction oscillator with stiction, introducing a new concept of stiction solutions, analyzing non-uniqueness, and using regularization to reveal canard trajectories and periodic orbits that connect slip-stick behaviors.

## Contribution

The paper introduces the concept of stiction solutions for non-Filippov friction models, and uses regularization to analyze non-uniqueness and canard phenomena in the system.

## Key findings

- Identification of a repelling slow manifold separating slip and stick solutions.
- Discovery of canard trajectories delaying slip onset.
- Existence of a family of periodic orbits with saddle stability connecting slip-stick branches.

## Abstract

We study the solutions of a friction oscillator subject to stiction. This discontinuous model is non-Filippov, and the concept of Filippov solution cannot be used. Furthermore some Carath\'eodory solutions are unphysical. Therefore we introduce the concept of stiction solutions: these are the Carath\'eodory solutions that are physically relevant, i.e. the ones that follow the stiction law. However, we find that some of the stiction solutions are forward non-unique in subregions of the slip onset. We call these solutions singular, in contrast to the regular stiction solutions that are forward unique. In order to further the understanding of the non-unique dynamics, we introduce a regularization of the model. This gives a singularly perturbed problem that captures the main features of the original discontinuous problem. We identify a repelling slow manifold that separates the forward slipping to forward sticking solutions, leading to a high sensitivity to the initial conditions. On this slow manifold we find canard trajectories, that have the physical interpretation of delaying the slip onset. We show with numerics that the regularized problem has a family of periodic orbits interacting with the canards. We observe that this family has a saddle stability and that it connects, in the rigid body limit, the two regular, slip-stick branches of the discontinuous problem, that were otherwise disconnected.