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

Research output: Contribution to journalJournal article – Annual report year: 2018Researchpeer-review

Standard

Canards in stiction: on solutions of a friction oscillator by regularization. / Bossolini, Elena; Brøns, Morten; Kristiansen, Kristian Uldall.

In: S I A M Journal on Applied Dynamical Systems, Vol. 16, No. 4, 2017, p. 2233–2258.

Research output: Contribution to journalJournal article – Annual report year: 2018Researchpeer-review

Harvard

APA

CBE

MLA

Vancouver

Author

Bibtex

@article{bba44d0ae3dc4e4996e436968316b985,
title = "Canards in stiction: on solutions of a friction oscillator by regularization",
abstract = "We study the solutions of a friction oscillator subject to stiction. This discontinuous model is nonFilippov, 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.",
keywords = "Stiction, Friction oscillator, non-Filippov, Regularization, Canard, Slip-stick, Delayed slip onset",
author = "Elena Bossolini and Morten Br{\o}ns and Kristiansen, {Kristian Uldall}",
year = "2017",
doi = "10.1137/17M1120774",
language = "English",
volume = "16",
pages = "2233–2258",
journal = "S I A M Journal on Applied Dynamical Systems",
issn = "1536-0040",
publisher = "Society for Industrial and Applied Mathematics",
number = "4",

}

RIS

TY - JOUR

T1 - Canards in stiction: on solutions of a friction oscillator by regularization

AU - Bossolini, Elena

AU - Brøns, Morten

AU - Kristiansen, Kristian Uldall

PY - 2017

Y1 - 2017

N2 - We study the solutions of a friction oscillator subject to stiction. This discontinuous model is nonFilippov, 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.

AB - We study the solutions of a friction oscillator subject to stiction. This discontinuous model is nonFilippov, 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.

KW - Stiction

KW - Friction oscillator

KW - non-Filippov

KW - Regularization

KW - Canard

KW - Slip-stick

KW - Delayed slip onset

U2 - 10.1137/17M1120774

DO - 10.1137/17M1120774

M3 - Journal article

VL - 16

SP - 2233

EP - 2258

JO - S I A M Journal on Applied Dynamical Systems

JF - S I A M Journal on Applied Dynamical Systems

SN - 1536-0040

IS - 4

ER -