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Set-oriented numerical computation of rotation sets

Publikation: Beiträge in Zeitschriften › Zeitschriftenaufsätze › Forschung › begutachtet

Standard

Set-oriented numerical computation of rotation sets. / Polotzek, Katja; Padberg-Gehle, Kathrin; Oertel-Jäger, Tobias.
in: Journal of Computational Dynamics, Jahrgang 4, Nr. 1, 01.11.2017, S. 119-141.

Publikation: Beiträge in Zeitschriften › Zeitschriftenaufsätze › Forschung › begutachtet

Harvard

Polotzek, K, Padberg-Gehle, K & Oertel-Jäger, T 2017, 'Set-oriented numerical computation of rotation sets', Journal of Computational Dynamics, Jg. 4, Nr. 1, S. 119-141. https://doi.org/10.3934/jcd.2017004

APA

Polotzek, K., Padberg-Gehle, K., & Oertel-Jäger, T. (2017). Set-oriented numerical computation of rotation sets. Journal of Computational Dynamics, 4(1), 119-141. https://doi.org/10.3934/jcd.2017004

Vancouver

Polotzek K, Padberg-Gehle K, Oertel-Jäger T. Set-oriented numerical computation of rotation sets. Journal of Computational Dynamics. 2017 Nov 1;4(1):119-141. doi: 10.3934/jcd.2017004

Bibtex

@article{5e5a7dd2cc0041d6a8de0da7296347c8,
title = "Set-oriented numerical computation of rotation sets",
abstract = "We establish a set-oriented algorithm for the numerical approximationof the rotation set of homeomorphisms of the two-torus homotopic tothe identity. A theoretical background is given by the concept of ε-rotationsets. These are obtained by replacing orbits with ε-pseudo-orbits in the definitionof the Misiurewicz-Ziemian rotation set and are shown to converge to thelatter as ε decreases to zero. Based on this result, we prove the convergenceof the numerical approximations as precision and iteration time tend to infinity.Further, we provide analytic error estimates for the algorithm under anadditional boundedness assumption, which is known to hold in many relevantcases and in particular for non-empty interior rotation sets.",
keywords = "Mathematics, Rotation Theory, rotation sets, pseudo-orbits, set-oriented numerics",
author = "Katja Polotzek and Kathrin Padberg-Gehle and Tobias Oertel-J{\"a}ger",
note = "Publisher Copyright: {\textcopyright} American Institute of Mathematical Sciences.",
year = "2017",
month = nov,
day = "1",
doi = "10.3934/jcd.2017004",
language = "English",
volume = "4",
pages = "119--141",
journal = "Journal of Computational Dynamics",
issn = "2158-2491",
publisher = "American Institute of Mathematical Sciences - AIMS Press",
number = "1",

}

RIS

TY - JOUR

T1 - Set-oriented numerical computation of rotation sets

AU - Polotzek, Katja

AU - Padberg-Gehle, Kathrin

AU - Oertel-Jäger, Tobias

N1 - Publisher Copyright: © American Institute of Mathematical Sciences.

PY - 2017/11/1

Y1 - 2017/11/1

N2 - We establish a set-oriented algorithm for the numerical approximationof the rotation set of homeomorphisms of the two-torus homotopic tothe identity. A theoretical background is given by the concept of ε-rotationsets. These are obtained by replacing orbits with ε-pseudo-orbits in the definitionof the Misiurewicz-Ziemian rotation set and are shown to converge to thelatter as ε decreases to zero. Based on this result, we prove the convergenceof the numerical approximations as precision and iteration time tend to infinity.Further, we provide analytic error estimates for the algorithm under anadditional boundedness assumption, which is known to hold in many relevantcases and in particular for non-empty interior rotation sets.

AB - We establish a set-oriented algorithm for the numerical approximationof the rotation set of homeomorphisms of the two-torus homotopic tothe identity. A theoretical background is given by the concept of ε-rotationsets. These are obtained by replacing orbits with ε-pseudo-orbits in the definitionof the Misiurewicz-Ziemian rotation set and are shown to converge to thelatter as ε decreases to zero. Based on this result, we prove the convergenceof the numerical approximations as precision and iteration time tend to infinity.Further, we provide analytic error estimates for the algorithm under anadditional boundedness assumption, which is known to hold in many relevantcases and in particular for non-empty interior rotation sets.

KW - Mathematics

KW - Rotation Theory

KW - rotation sets

KW - pseudo-orbits

KW - set-oriented numerics

UR - http://www.scopus.com/inward/record.url?scp=85040731579&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/baa4c899-3756-3249-b150-8a6f1cbd87c0/

U2 - 10.3934/jcd.2017004

DO - 10.3934/jcd.2017004

M3 - Journal articles

VL - 4

SP - 119

EP - 141

JO - Journal of Computational Dynamics

JF - Journal of Computational Dynamics

SN - 2158-2491

IS - 1

ER -

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