A numerical approximation of the rotation number
type: Item
instance of: scholarly article
title: A numerical approximation of the rotation number
zbMATH Open document ID: 0840.65066
DOI: 10.1016/0096-3003(94)00249-5
published in: Applied Mathematics and Computation
publication date: 1996-06-30T00:00:00Z
review text: This paper is concerned with the numerical computation of the rotation number of homeomorphisms \(f : S^1 \to S^1\) on the circle. As it is well known this number allows to obtain important properties of the orbit defined by \(f\), thus it is a rational number if and only if \(f\) defines a periodic orbit and if it is irrational the orbit is dense on \(S^1\). Moreover the study of rotation numbers of a parameter dependent family of diffeomorphisms allows to explain the dynamical behaviour of bifurcations [see e.g. \textit{J. Guckenheimer} and \textit{P. Holmes}, Nonlinear oscillations, dynamical systems and bifurcations of vector fields (1983; Zbl 0515.34001)]. The author proves a theorem which states some computable conditions that allow to determine intervals of existence of the rotation number in the case that it is an irrational number and its exact value in the rational case. As it is expected, smaller intervals of existence of the rotation numbers can be obtained at the price to increase the computational cost. Some numerical experiments with maps on the circle which are known to possess rational and irrational rotation numbers are presented to show the behaviour of the results given in the theorem. Finally, the same algorithm is applied to a family of two-dimensional maps, the delayed logistic map, and it is found that also in this case the rotation numbers are computed with remarkable accuracy.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 37E99
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 847554
zbMATH Keywords: rotation number
zbMATH Keywords: periodic orbit
zbMATH Keywords: bifurcations
zbMATH Keywords: numerical experiments
author: Raffaella Pavani
reviewed by: Manuel Calvo
MaRDI profile type: Publication
full work available at URL: 0096-3003(94)00249-5
OpenAlex ID: W1991567631
cites work: Q3997135
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: The homogenization of an ordinary differential equation with a chessboard structure
cites work: Conjugating the Poincaré-map to a rotation
cites work: Q5564439
cites work: Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer-assisted study
cites work: On the numerical approximation of the rotation number
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description: scientific article; zbMATH DE number 847554