Self-rotation number using the turning angle
type: Item
instance of: scholarly article
title: Self-rotation number using the turning angle
zbMATH Open document ID: 0969.37019
DOI: 10.1016/S0167-2789(00)00093-2
author: D. G. Sterling
author: H. R. Dullin
author: J. D. Meiss
published in: Physica D
publication date: 2001-09-25T00:00:00Z
review text: For circle homeomorphisms or monotone twist area preserving maps on the annulus \(\mathbb{S}^1\times\mathbb{R}\), the notion of rotation number of an orbit is well known. If the map \(F\) is defined on \(\mathbb{R}^2\), and \(x^*\) is a fixed point for \(F\), then we can define the rotation number of the orbit \({\mathcal O}(x)\) as the rotation number of the point \(x\) associated to the map \(F\) restricted to \(\mathbb{R}^2\setminus\{x^*\}\cong\mathbb{S}^1\times\mathbb{R}^+\). In this case the rotation number measures the average rate of rotation around \(x^{*}\). This definition has a drawback: the rotation number can change discontinuously as the parameter in the map's definition varies continuously, even when no bifurcations occur. In [\textit{B. B. Peckham}, Nonlinearity 3, No. 2, 261-280 (1990; Zbl 0704.58035)] the self-rotation number of a periodic orbit of an homeomorphism of \(\mathbb{R}^{2}\) is defined, as an alternative notion of rotation number. In order to simplify the computation of the self-rotation number, the paper under review proposes a construction of this number based on the so called turning angle. The turning angle of an orbit of the map \(F\) measures the average rate of turning of the vector \(V=F(y)-y\) along the orbit \({\mathcal O}(x)\). Let \(\varphi:\mathbb{R}^{2}\to\mathbb{S}^{1}\) be the function that associates to a point \(x\in\mathbb{R}^{2}\) its turning angle \(\theta\in\mathbb{S}^{1}\). Then the self-rotation number of \(x\) is defined by: \[ \rho_s(x)=\lim_{n\to\infty}\sum_{i=1}^{n}\varphi(F^{i}(x)) \] The function \(\varphi\) is not surjective in general. If \(\alpha\not\in\text{Im}(\varphi)\), we say that the homeomorphism \(F\) avoids the angle \(\alpha\). It is proved that if a one parameter family \(f_\mu\) of the plane avoids the angle \(\alpha(\mu)\), then the self-rotation number \(\rho_{s}(x)\) is invariant under continuation of periodic orbits of \(f_\mu\). For example every orientation preserving nonsingular linear…
reviewed by: Emilia Petrisor
Mathematics Subject Classification ID: 37E45
Mathematics Subject Classification ID: 37E30
Mathematics Subject Classification ID: 37B10
zbMATH DE Number: 1530307
zbMATH Keywords: rotation number
zbMATH Keywords: self-rotation number
zbMATH Keywords: turning angle
zbMATH Keywords: Hénon map
zbMATH Keywords: anti-integrable limit
MaRDI profile type: Publication
full work available at URL: s0167-2789(00)00093-2
OpenAlex ID: W2090957015
description: scientific article; zbMATH DE number 1530307