A numerical study of the likelihood of phase locking
type: Item
instance of: scholarly article
title: A numerical study of the likelihood of phase locking
zbMATH Open document ID: 0598.58041
DOI: 10.1016/0167-2789(85)90099-5
author: Oscar E. III Lanford
published in: Physica D
publication date: 1985-01-01T00:00:00Z
review text: Let \(f\) be a continuous strictly increasing mapping of the real line to itself \((f(x+1)=f(x)+1)\). By passing to quotients one can obtain a continuous one-to-one mapping \(\tilde f\) of the circle -- identified with the real numbers modulo the integers -- to itself. An orientation-preserving homeomorphism of the circle can be defined to be a mapping obtained in this way. The mapping \(f\) is not uniquely determined by the induced mapping \(\tilde f\); two \(f\)'s induce the same \(\tilde f\) iff they differ by an integer constant. From the dynamical point of view, orientation-preserving homeomorphisms of the circle can be grouped into two classes, those which have at least one periodic cycle and those which do not. A mapping with at least one cycle is said to be phase locking. The author's main purpose is to study the two-parameter family of mappings \[ f_{K,\Omega}(x)=x+\Omega +(K/2\pi)\sin (2\pi x) \] where \(\Omega\) lie in \([0,1)\) and \(| K| \leq 1\) (the last inequality implies: \(f_{K,\Omega}\) is nondecreasing). So the author makes a numerical computation of the size of the set of \(\Omega\)'s for which the homeomorphism of the circle \(\tilde f_{K,\Omega}(x)=x+\Omega +(K/2\pi)\sin (2\pi x) \bmod 1\) is phase locked. The obtained numerical results confirm Arnold's picture, that is, from the topological standpoint, phase locking is generic (cf. \textit{V. I. Arnol'd} [Am. Math. Soc., Transl., II. Ser. 46, 213--284 (1965; Zbl 0152.41905); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 25, 21--86 (1961; Zbl 0135.42603)]).
reviewed by: Vasil G. Angelov
Mathematics Subject Classification ID: 37E10
zbMATH DE Number: 3965049
zbMATH Keywords: phase locking
zbMATH Keywords: numerical computation
zbMATH Keywords: orientation-preserving homeomorphisms of the circle
zbMATH Keywords: phase locking is generic
MaRDI profile type: Publication
full work available at URL: 0167-2789(85)90099-5
OpenAlex ID: W2070312520
cites work: Small denominators. I. Mappings of the circumference onto itself
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cites work: Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. (On smooth conjugacy of diffeomorphisms of the circle with rotations)
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description: scientific article; zbMATH DE number 3965049