Periodic orbits of the sawtooth maps
type: Item
instance of: scholarly article
title: Periodic orbits of the sawtooth maps
zbMATH Open document ID: 0648.58027
DOI: 10.1016/0167-2789(88)90104-2
author: Nigel Bird
author: Franco Vivaldi
published in: Physica D
publication date: 1988-01-01T00:00:00Z
review text: The sawtooth maps are a one-parameter family of chaotic area-preserving maps of the 2-torus, which become Anosov systems for special values of the parameter. The purpose of this paper is to carry out a systematic study of the periodic orbits of the sawtooth maps, based on the knowledge of Anosov orbits. Since the sawtooth maps are piecewise linear, the authors of this paper are able to construct the parameter dependence of the sawtooth orbits (perturbed Anosov orbits) explicitly in terms of rational functions with integer coefficients. The essential ingredients for this method are arithmetic and symbolic dynamics. Arithmetical methods are used to compute the unperturbed (i.e., Anosov) orbits, which are then coded as symbol sequences. The latter are then used to classify periodic orbits and to construct explicitly their parameter dependence.
reviewed by: Ding Tongren
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 4058481
zbMATH Keywords: sawtooth maps
zbMATH Keywords: chaos
zbMATH Keywords: chaotic area-preserving maps
zbMATH Keywords: Anosov systems
zbMATH Keywords: periodic orbits
zbMATH Keywords: perturbed Anosov orbits
zbMATH Keywords: arithmetic and symbolic dynamics
MaRDI profile type: Publication
full work available at URL: 0167-2789(88)90104-2
OpenAlex ID: W2050909018
cites work: Newton method for highly unstable orbits
cites work: Q3998228
cites work: Q5608683
cites work: Arithmetical properties of strongly chaotic motions
cites work: A linear code for the sawtooth and cat maps
cites work: Transport in Hamiltonian systems
cites work: A variational principle for invariant tori of fixed frequency
cites work: Regular and stochastic motion
cites work: Q3867858
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description: scientific article; zbMATH DE number 4058481