Bifurcations of circle maps: Arnol'd tongues, bistability and rotation intervals
type: Item
instance of: scholarly article
title: Bifurcations of circle maps: Arnol'd tongues, bistability and rotation intervals
zbMATH Open document ID: 0612.58032
DOI: 10.1007/BF01207252
author: Philip Boyland
published in: Communications in Mathematical Physics
publication date: 1986-01-01T00:00:00Z
review text: This paper studies the bifurcations of families of noninvertible circle maps defined by \(x+w+bp(x)\) (mod 1), where (b,w) are two real parameters, p(x) being a continuous periodic function of period 1, which verifies other hypotheses given in {\S} 3. When \(p(x)\equiv (2\pi)^{-1}\sin (2\pi x)\), a ''standard'' map is obtained. More particularly, the author considers the bifurcation curves in the parameter plane (b,w), which are the boundaries of the existence domains of periodic points associated with a rational rotation number. In this case, such domains, with \(b>1\), are the extension (with overlappings) of what he calls ''Arnol'd tongues'' (known before Arnol'd in the theory of nonlinear harmonic and subharmonic synchronization of oscillators), obtained when the map is a diffeomorphism \((| b| <1\) and not \(b<1\) as indicated in the text). The case corresponding to irrational rotation numbers is also considered. Theorems give information about the dynamics (bistability and aperiodicity) of the maps for (b,w) belonging to various regions of the parameter plane.
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 57R35
Mathematics Subject Classification ID: 70K50
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 3990452
zbMATH Keywords: circle endomorphism
zbMATH Keywords: periodic points
zbMATH Keywords: transition to chaos
zbMATH Keywords: rotation numbers
zbMATH Keywords: bifurcations
zbMATH Keywords: noninvertible circle maps
reviewed by: Christian Mira
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3990452