Scaling exponents, relations, and order dependence for circle maps
type: Item
instance of: scholarly article
title: Scaling exponents, relations, and order dependence for circle maps
zbMATH Open document ID: 0612.58022
DOI: 10.1016/0167-2789(87)90233-8
author: D. R. Rasmussen
author: Preben Alstrøm
author: M. T. Levinsen
published in: Physica D
publication date: 1987-01-01T00:00:00Z
review text: Scaling relations describing devil's staircase structures obtained by iterations of orientation-preserving circle maps with minimal slope converging to zero have been investigated. In the limit, the circle map has a zero slope inflection point and the influence of the order at this point on the scaling exponents is analyzed. We study the behavior of the step sizes and the minimal distance to an integer from a (non-zero) cycle point for the super-stable orbit, and in particular we treat the average behavior. This leads to a definition of an ''average crossover scale exponent'' \({\bar \nu}\) and an ''average measure exponent'' \({\bar \beta}\), besides the earlier defined ''average dimension'' \(\bar D.\) We find \({\bar \nu}\cong \nu\), \({\bar \beta}\cong \beta\), and \(\bar D\cong D\). Moreover, \({\bar \beta}=(1-\bar D){\bar \nu}\). Finally, as the order increases we observe that hourglass structures evolve in the Arnold tongues.
Mathematics Subject Classification ID: 37B99
Mathematics Subject Classification ID: 26A18
zbMATH DE Number: 3990442
zbMATH Keywords: Scaling
zbMATH Keywords: devil's staircase
zbMATH Keywords: orientation-preserving circle maps
MaRDI profile type: Publication
full work available at URL: 0167-2789(87)90233-8
OpenAlex ID: W2062627882
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description: scientific article; zbMATH DE number 3990442